tom leinster university of edinburghqpl2016.cis.strath.ac.uk/pdfs/1leinster.pdflinear algebra done...
TRANSCRIPT
![Page 1: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/1.jpg)
In search of the spectrum
Tom Leinster
University of Edinburgh
![Page 2: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/2.jpg)
Meanings of ‘spectrum’
Physics & chemistry: emission/absorption spectra, mass spectroscopy, etc.
Linear algebra: eigenvalues of an operator, with their algebraic multiplicities
Functional analysis and PDEs: more sophisticated relatives of the linearalgebra notion
Commutative algebra: the spectrum of a commutative ring is its space ofprime ideals
. . . and many more meanings, all related to one another.
![Page 3: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/3.jpg)
Meanings of ‘spectrum’Physics & chemistry: emission/absorption spectra, mass spectroscopy, etc.
Linear algebra: eigenvalues of an operator, with their algebraic multiplicities
Functional analysis and PDEs: more sophisticated relatives of the linearalgebra notion
Commutative algebra: the spectrum of a commutative ring is its space ofprime ideals
. . . and many more meanings, all related to one another.
![Page 4: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/4.jpg)
Meanings of ‘spectrum’Physics & chemistry: emission/absorption spectra, mass spectroscopy, etc.
Linear algebra: eigenvalues of an operator, with their algebraic multiplicities
Functional analysis and PDEs: more sophisticated relatives of the linearalgebra notion
Commutative algebra: the spectrum of a commutative ring is its space ofprime ideals
. . . and many more meanings, all related to one another.
![Page 5: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/5.jpg)
Meanings of ‘spectrum’Physics & chemistry: emission/absorption spectra, mass spectroscopy, etc.
Linear algebra: eigenvalues of an operator, with their algebraic multiplicities
Functional analysis and PDEs: more sophisticated relatives of the linearalgebra notion
Commutative algebra: the spectrum of a commutative ring is its space ofprime ideals
. . . and many more meanings, all related to one another.
![Page 6: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/6.jpg)
Meanings of ‘spectrum’Physics & chemistry: emission/absorption spectra, mass spectroscopy, etc.
Linear algebra: eigenvalues of an operator, with their algebraic multiplicities
Functional analysis and PDEs: more sophisticated relatives of the linearalgebra notion
Commutative algebra: the spectrum of a commutative ring is its space ofprime ideals
. . . and many more meanings, all related to one another.
![Page 7: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/7.jpg)
Meanings of ‘spectrum’Physics & chemistry: emission/absorption spectra, mass spectroscopy, etc.
Linear algebra: eigenvalues of an operator, with their algebraic multiplicities
Functional analysis and PDEs: more sophisticated relatives of the linearalgebra notion
Commutative algebra: the spectrum of a commutative ring is its space ofprime ideals
. . . and many more meanings, all related to one another.
![Page 8: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/8.jpg)
Unrelated meanings of ‘spectrum’
Homotopy theory: spectrum ≈ infinite loop space.
Non-quantum computing:
![Page 9: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/9.jpg)
Unrelated meanings of ‘spectrum’Homotopy theory: spectrum ≈ infinite loop space.
Non-quantum computing:
![Page 10: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/10.jpg)
Unrelated meanings of ‘spectrum’Homotopy theory: spectrum ≈ infinite loop space.
Non-quantum computing:
![Page 11: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/11.jpg)
Unrelated meanings of ‘spectrum’Homotopy theory: spectrum ≈ infinite loop space.
Non-quantum computing:
![Page 12: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/12.jpg)
What ‘spectrum’ means in this talk
I will use ‘spectrum’ to mean the set Spec(T ) of eigenvaluesof a linear operator T on a finite-dimensional vector space,with their algebraic multiplicities.
Algebraically and categorically, its properties seem awkward.
For instance, given operators S and T on the same space,knowing Spec(S) and Spec(T ) tells you almost nothing aboutSpec(S T ) or Spec(S +T ).
But socially, the spectrum is important!
So, there ought to be a clean abstract characterization of it.
This talk offers one.
![Page 13: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/13.jpg)
What ‘spectrum’ means in this talk
I will use ‘spectrum’ to mean the set Spec(T ) of eigenvaluesof a linear operator T on a finite-dimensional vector space,with their algebraic multiplicities.
Algebraically and categorically, its properties seem awkward.
For instance, given operators S and T on the same space,knowing Spec(S) and Spec(T ) tells you almost nothing aboutSpec(S T ) or Spec(S +T ).
But socially, the spectrum is important!
So, there ought to be a clean abstract characterization of it.
This talk offers one.
![Page 14: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/14.jpg)
What ‘spectrum’ means in this talk
I will use ‘spectrum’ to mean the set Spec(T ) of eigenvaluesof a linear operator T on a finite-dimensional vector space,with their algebraic multiplicities.
Algebraically and categorically, its properties seem awkward.
For instance, given operators S and T on the same space,knowing Spec(S) and Spec(T ) tells you almost nothing aboutSpec(S T ) or Spec(S +T ).
But socially, the spectrum is important!
So, there ought to be a clean abstract characterization of it.
This talk offers one.
![Page 15: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/15.jpg)
What ‘spectrum’ means in this talk
I will use ‘spectrum’ to mean the set Spec(T ) of eigenvaluesof a linear operator T on a finite-dimensional vector space,with their algebraic multiplicities.
Algebraically and categorically, its properties seem awkward.
For instance, given operators S and T on the same space,knowing Spec(S) and Spec(T ) tells you almost nothing aboutSpec(S T ) or Spec(S +T ).
But socially, the spectrum is important!
So, there ought to be a clean abstract characterization of it.
This talk offers one.
![Page 16: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/16.jpg)
What ‘spectrum’ means in this talk
I will use ‘spectrum’ to mean the set Spec(T ) of eigenvaluesof a linear operator T on a finite-dimensional vector space,with their algebraic multiplicities.
Algebraically and categorically, its properties seem awkward.
For instance, given operators S and T on the same space,knowing Spec(S) and Spec(T ) tells you almost nothing aboutSpec(S T ) or Spec(S +T ).
But socially, the spectrum is important!
So, there ought to be a clean abstract characterization of it.
This talk offers one.
![Page 17: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/17.jpg)
What ‘spectrum’ means in this talk
I will use ‘spectrum’ to mean the set Spec(T ) of eigenvaluesof a linear operator T on a finite-dimensional vector space,with their algebraic multiplicities.
Algebraically and categorically, its properties seem awkward.
For instance, given operators S and T on the same space,knowing Spec(S) and Spec(T ) tells you almost nothing aboutSpec(S T ) or Spec(S +T ).
But socially, the spectrum is important!
So, there ought to be a clean abstract characterization of it.
This talk offers one.
![Page 18: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/18.jpg)
What ‘spectrum’ means in this talk
I will use ‘spectrum’ to mean the set Spec(T ) of eigenvaluesof a linear operator T on a finite-dimensional vector space,with their algebraic multiplicities.
Algebraically and categorically, its properties seem awkward.
For instance, given operators S and T on the same space,knowing Spec(S) and Spec(T ) tells you almost nothing aboutSpec(S T ) or Spec(S +T ).
But socially, the spectrum is important!
So, there ought to be a clean abstract characterization of it.
This talk offers one.
![Page 19: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/19.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 20: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/20.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 21: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/21.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,
the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 22: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/22.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariant
is the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 23: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/23.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariant
is the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 24: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/24.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 25: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/25.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 26: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/26.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 27: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/27.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 28: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/28.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 29: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/29.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 30: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/30.jpg)
Outline
The goal of the talk is to explain this theorem:
Theorem
Among all invariants of linear operators on finite-dimensional vector spaces,the universal cyclic, balanced invariantis the set of nonzero eigenvalues with their algebraic multiplicities.
Plan:
1. Linear Algebra Done Right
2. Invariants
3. Cyclic invariants
4. Balanced invariants
5. The theorem
![Page 31: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/31.jpg)
1. Linear Algebra Done Right
![Page 32: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/32.jpg)
Linear Algebra Done Rightby Sheldon Axler
Sheldon Axler (1975)
Book (1996)
![Page 33: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/33.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 34: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/34.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 35: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/35.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 36: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/36.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 37: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/37.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 38: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/38.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 39: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/39.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 40: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/40.jpg)
The eventual image and eventual kernel
Throughout:
let k be an algebraically closed field
let X be a finite-dimensional vector space over k
let T be a linear operator on X (i.e. a linear map T ∶X Ð→ X ).
The eventual image im∞(T ) of T is the intersection of the chain ofsubspaces
im(T ) ⊇ im(T 2) ⊇ im(T 3) ⊇ ⋯.
This is smaller than the ordinary image im(T ).
The eventual kernel ker∞(T ) of T is the union of the chain of subspaces
ker(T ) ⊆ ker(T 2) ⊆ ker(T 3) ⊆ ⋯.
This is larger than the ordinary kernel ker(T ).
![Page 41: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/41.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and (ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 42: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/42.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and (ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 43: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/43.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and (ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 44: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/44.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and (ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 45: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/45.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and
(ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 46: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/46.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and (ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 47: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/47.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and (ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 48: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/48.jpg)
The first canonical decomposition of a linear operator
Recall the notation: T is a linear operator on a fin-dim vector space X .
Lemma
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent (i.e. some power of T0 is 0) and T× is invertible.
In other words: (i) X = ker∞(T )⊕ im∞(T ), and (ii) T restricts to anilpotent operator on ker∞(T ) and an invertible operator on im∞(T ).
This is the unique decomposition of T as the direct sum of a nilpotentoperator and an invertible operator.
Contrast: usually X ≠ ker(T ) + im(T ), although we do havedim X = dim ker(T ) + dim im(T ).
![Page 49: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/49.jpg)
The second canonical decomposition of a linear operator
Theorem
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent (i.e. the only eigenvalue of Tλ is λ).
We call ker∞(T − λ) the eventual eigenspace (or generalized eigenspace)with value λ.
It is bigger than the ordinary eigenspace, which consists of those vectorsannihilated by applying T − λ just once.
Contrast: usually X ≠⊕λ∈k
ker(T − λ). They are equal iff T is diagonalizable.
The eventual eigenspace ker∞(T − λ) is trivial for all except finitely manyvalues of λ ∈ k — namely, the eigenvalues.
![Page 50: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/50.jpg)
The second canonical decomposition of a linear operator
Theorem
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent (i.e. the only eigenvalue of Tλ is λ).
We call ker∞(T − λ) the eventual eigenspace (or generalized eigenspace)with value λ.
It is bigger than the ordinary eigenspace, which consists of those vectorsannihilated by applying T − λ just once.
Contrast: usually X ≠⊕λ∈k
ker(T − λ). They are equal iff T is diagonalizable.
The eventual eigenspace ker∞(T − λ) is trivial for all except finitely manyvalues of λ ∈ k — namely, the eigenvalues.
![Page 51: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/51.jpg)
The second canonical decomposition of a linear operator
Theorem
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent (i.e. the only eigenvalue of Tλ is λ).
We call ker∞(T − λ) the eventual eigenspace (or generalized eigenspace)with value λ.
It is bigger than the ordinary eigenspace, which consists of those vectorsannihilated by applying T − λ just once.
Contrast: usually X ≠⊕λ∈k
ker(T − λ). They are equal iff T is diagonalizable.
The eventual eigenspace ker∞(T − λ) is trivial for all except finitely manyvalues of λ ∈ k — namely, the eigenvalues.
![Page 52: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/52.jpg)
The second canonical decomposition of a linear operator
Theorem
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent (i.e. the only eigenvalue of Tλ is λ).
We call ker∞(T − λ) the eventual eigenspace (or generalized eigenspace)with value λ.
It is bigger than the ordinary eigenspace, which consists of those vectorsannihilated by applying T − λ just once.
Contrast: usually X ≠⊕λ∈k
ker(T − λ). They are equal iff T is diagonalizable.
The eventual eigenspace ker∞(T − λ) is trivial for all except finitely manyvalues of λ ∈ k — namely, the eigenvalues.
![Page 53: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/53.jpg)
The second canonical decomposition of a linear operator
Theorem
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent (i.e. the only eigenvalue of Tλ is λ).
We call ker∞(T − λ) the eventual eigenspace (or generalized eigenspace)with value λ.
It is bigger than the ordinary eigenspace, which consists of those vectorsannihilated by applying T − λ just once.
Contrast: usually X ≠⊕λ∈k
ker(T − λ). They are equal iff T is diagonalizable.
The eventual eigenspace ker∞(T − λ) is trivial for all except finitely manyvalues of λ ∈ k — namely, the eigenvalues.
![Page 54: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/54.jpg)
The second canonical decomposition of a linear operator
Theorem
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent (i.e. the only eigenvalue of Tλ is λ).
We call ker∞(T − λ) the eventual eigenspace (or generalized eigenspace)with value λ.
It is bigger than the ordinary eigenspace, which consists of those vectorsannihilated by applying T − λ just once.
Contrast: usually X ≠⊕λ∈k
ker(T − λ). They are equal iff T is diagonalizable.
The eventual eigenspace ker∞(T − λ) is trivial for all except finitely manyvalues of λ ∈ k — namely, the eigenvalues.
![Page 55: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/55.jpg)
The second canonical decomposition of a linear operator
Theorem
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent (i.e. the only eigenvalue of Tλ is λ).
We call ker∞(T − λ) the eventual eigenspace (or generalized eigenspace)with value λ.
It is bigger than the ordinary eigenspace, which consists of those vectorsannihilated by applying T − λ just once.
Contrast: usually X ≠⊕λ∈k
ker(T − λ). They are equal iff T is diagonalizable.
The eventual eigenspace ker∞(T − λ) is trivial for all except finitely manyvalues of λ ∈ k — namely, the eigenvalues.
![Page 56: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/56.jpg)
Comparing the two decompositions
First decomposition:
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent and T× is invertible.
Second decomposition:
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent.
The two operators called ‘T0’ are the same, and the second decompositionrefines the first:
im∞(T )⤾T× =⊕λ≠0
ker∞(T − λ)⤾Tλ .
![Page 57: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/57.jpg)
Comparing the two decompositions
First decomposition:
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent and T× is invertible.
Second decomposition:
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent.
The two operators called ‘T0’ are the same, and the second decompositionrefines the first:
im∞(T )⤾T× =⊕λ≠0
ker∞(T − λ)⤾Tλ .
![Page 58: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/58.jpg)
Comparing the two decompositions
First decomposition:
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent and T× is invertible.
Second decomposition:
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent.
The two operators called ‘T0’ are the same, and the second decompositionrefines the first:
im∞(T )⤾T× =⊕λ≠0
ker∞(T − λ)⤾Tλ .
![Page 59: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/59.jpg)
Comparing the two decompositions
First decomposition:
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent and T× is invertible.
Second decomposition:
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent.
The two operators called ‘T0’ are the same
, and the second decompositionrefines the first:
im∞(T )⤾T× =⊕λ≠0
ker∞(T − λ)⤾Tλ .
![Page 60: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/60.jpg)
Comparing the two decompositions
First decomposition:
X⤾T = ker∞(T )⤾T0 ⊕ im∞(T )⤾T×
where T0 is nilpotent and T× is invertible.
Second decomposition:
X⤾T =⊕λ∈k
ker∞(T − λ)⤾Tλ
where Tλ − λ is nilpotent.
The two operators called ‘T0’ are the same, and the second decompositionrefines the first:
im∞(T )⤾T× =⊕λ≠0
ker∞(T − λ)⤾Tλ .
![Page 61: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/61.jpg)
Algebraic multiplicity
Let λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 62: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/62.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 63: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/63.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 64: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/64.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 65: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/65.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 66: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/66.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 67: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/67.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 68: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/68.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 69: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/69.jpg)
Algebraic multiplicityLet λ ∈ k .
The algebraic multiplicity of λ in T is
αT (λ) = dim ker∞(T − λ).
(We could also call it the ‘dynamic multiplicity’.)
Note thatX =⊕
λ∈kker∞(T − λ) ⇒ dim X = ∑
λ∈kαT (λ).
We can then define:
Trace: tr(T ) = ∑λ∈k
αT (λ) ⋅ λ
Determinant: det(T ) =∏λ∈k
λαT (λ)
Characteristic polynomial: χT (x) =∏λ∈k
(x − λ)αT (λ) = det(x −T ).
All have their usual meanings!
![Page 70: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/70.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 71: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/71.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 72: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/72.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 73: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/73.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 74: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/74.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 75: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/75.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 76: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/76.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 77: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/77.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction:
any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 78: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/78.jpg)
Functoriality of the eventual image
Given a category C , let Endo(C ) denote the category whose:
objects are endomorphisms X⤾T in C
maps X⤾T Ð→ Y⤾S are maps f ∶X Ð→ Y in C such that S f = f T .
Let FDVect be the category of finite-dimensional vector spaces.
We’re interested in Endo(FDVect), the category of linear operators.
There is a functor
Endo(FDVect) Ð→ Endo(FDVect)X⤾T ↦ im∞(T )⤾T× .
On maps, it’s defined by restriction: any map of operators f ∶X⤾T Ð→ Y⤾S
restricts to a map f ∶ im∞(T )⤾T× Ð→ im∞(S)⤾S× .
![Page 79: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/79.jpg)
2. Invariants
![Page 80: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/80.jpg)
Definition and examples
Let E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 81: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/81.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 82: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/82.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 83: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/83.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 84: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/84.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 85: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/85.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 86: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/86.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 87: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/87.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 88: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/88.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .
(Can describe these iso types concretely via Jordan normal form.)
![Page 89: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/89.jpg)
Definition and examplesLet E be a category. An invariant of objects of E is a function
ob (E )/≅ = isomorphism classes of objects of E Ð→ Ω,
where Ω is some set.
We’re studying invariants of linear operators. So take E = Endo(FDVect).
Examples of invariants of linear operators X⤾T :
The trace or determinant or characteristic polynomial.
The algebraic multiplicity αT (33) (etc.).
The spectrum Spec(T ), defined as the set of eigenvalues with theiralgebraic multiplicities. This is a finite subset-with-multiplicities of k.
The invertible spectrum Spec×(T ), defined as the set of nonzeroeigenvalues with their algebraic multiplicities. This is a finitesubset-with-multiplicities of k× = k ∖ 0.
The isomorphism type of im∞(T )⤾T× .(Can describe these iso types concretely via Jordan normal form.)
![Page 90: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/90.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T ), because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 91: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/91.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T ), because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 92: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/92.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T ), because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 93: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/93.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T ), because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 94: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/94.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T )
, because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 95: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/95.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T ), because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 96: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/96.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T ), because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 97: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/97.jpg)
Digression on the invertible spectrum
We just defined the invertible spectrum Spec×(T ) to be theset-with-multiplicities of nonzero eigenvalues.
There’s also Spec(T ), the set-with-multiplicities of all eigenvalues.
Suppose we know dim X .
Then knowing Spec×(T ) is equivalent to knowing Spec(T ), because
α0(T ) = dim X −∑λ≠0
αλ(T ).
Why are the nonzero eigenvalues interesting?
because they tell you the values of µ for which µT − I is singular
because of cyclicity. . .
![Page 98: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/98.jpg)
3. Cyclic invariants
![Page 99: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/99.jpg)
Definition
Let C be a category. An invariant Φ of endomorphisms in C is cyclic if
Φ(g f ) = Φ(f g)
whenever Xf //Yg
oo in C .
Example: Trace is cyclic: tr(g f ) = tr(f g).
A cyclic invariant Φ assigns a value to any cycle
Xn
X1
X2
X3
fnf1
f2
in C , since Φ(fi ⋯ f1 fn ⋯fi+1) is independent of i .
![Page 100: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/100.jpg)
DefinitionLet C be a category. An invariant Φ of endomorphisms in C is cyclic if
Φ(g f ) = Φ(f g)
whenever Xf //Yg
oo in C .
Example: Trace is cyclic: tr(g f ) = tr(f g).
A cyclic invariant Φ assigns a value to any cycle
Xn
X1
X2
X3
fnf1
f2
in C , since Φ(fi ⋯ f1 fn ⋯fi+1) is independent of i .
![Page 101: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/101.jpg)
DefinitionLet C be a category. An invariant Φ of endomorphisms in C is cyclic if
Φ(g f ) = Φ(f g)
whenever Xf //Yg
oo in C .
Example: Trace is cyclic: tr(g f ) = tr(f g).
A cyclic invariant Φ assigns a value to any cycle
Xn
X1
X2
X3
fnf1
f2
in C , since Φ(fi ⋯ f1 fn ⋯fi+1) is independent of i .
![Page 102: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/102.jpg)
DefinitionLet C be a category. An invariant Φ of endomorphisms in C is cyclic if
Φ(g f ) = Φ(f g)
whenever Xf //Yg
oo in C .
Example: Trace is cyclic: tr(g f ) = tr(f g).
A cyclic invariant Φ assigns a value to any cycle
Xn
X1
X2
X3
fnf1
f2
in C , since Φ(fi ⋯ f1 fn ⋯fi+1) is independent of i .
![Page 103: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/103.jpg)
The eventual image is a cyclic invariant
Given
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 104: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/104.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect
, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 105: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/105.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect)
, hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 106: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/106.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 107: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/107.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×
, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 108: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/108.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible
, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 109: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/109.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 110: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/110.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 111: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/111.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 112: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/112.jpg)
The eventual image is a cyclic invariantGiven
Xf //
Yg
oo
in FDVect, we get
X⤾gff //
Y⤾fgg
oo
in Endo(FDVect), hence by functoriality of the eventual image,
im∞(gf )⤾(gf )×f //
im∞(fg)⤾(fg)×g
oo
in Endo(FDVect).
But the composites off //g
oo are (gf )× and (fg)×, which are invertible, so
im∞(gf )⤾(gf )× ≅ im∞(fg)⤾(fg)× .
Conclusion: The isom’m type of im∞(T )⤾T× is a cyclic invariant of X⤾T .
(In fact, this is the initial cyclic invariant of linear operators.)
![Page 113: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/113.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 114: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/114.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 115: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/115.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 116: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/116.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 117: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/117.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 118: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/118.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 119: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/119.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 120: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/120.jpg)
The invertible spectrum is a cyclic invariant
Again, take Xf //Yg
oo in FDVect.
A similar argument shows that
ker∞(gf − λ) ≅ ker∞(fg − λ)
for all λ ≠ 0.
So gf and fg have the same nonzero eigenvalues with the same algebraicmultiplicities.
Conclusion: The invertible spectrum Spec× is a cyclic invariant.
Fact of life: This fails for the eigenvalue 0.
E.g. consider the first inclusion and first projection k k ⊕ k .One composite has 0 as an eigenvalue, and the other does not.
So, the multiplicity of 0 as an eigenvalue is not a cyclic invariant.
![Page 121: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/121.jpg)
4. Balanced invariants
![Page 122: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/122.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 123: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/123.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 124: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/124.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 125: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/125.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 126: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/126.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 127: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/127.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 128: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/128.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 129: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/129.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 130: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/130.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 131: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/131.jpg)
Looking at an object from all directions
Suppose someone tells you ‘left perfect’ is an important property of rings.
Then you know there must be an equally important invariant, ‘right perfect’.
Formally: given a ring R, define Rop by reversing the order of multiplication.
This defines an automorphism ( )op of the category of rings.
Then R is right perfect ⇐⇒ Rop is left perfect.
Given a compact metric space X , let N1(X ) be the number of balls ofradius 1 needed to cover X .
If N1(X ) is interesting then so too must be Nr(X ) (the number of balls ofradius r needed to cover X ), for every r > 0.
Formally: define tX to be X scaled up by a factor of t.
For each t > 0, this defines an automorphism t ⋅ − of the category of metricspaces.
Then Nr(X ) = N1((1/r)X ).
![Page 132: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/132.jpg)
Looking at an object from all directions
We agree that being invertible is an important property of linear operators.
For each λ ∈ k , we have an automorphism T ↦ T − λ of the categoryEndo(FDVect) of operators.
So, given an operator T , it must be important to ask for each λ ∈ k whetherT − λ is invertible. . . and of course it is important!
![Page 133: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/133.jpg)
Looking at an object from all directions
We agree that being invertible is an important property of linear operators.
For each λ ∈ k , we have an automorphism T ↦ T − λ of the categoryEndo(FDVect) of operators.
So, given an operator T , it must be important to ask for each λ ∈ k whetherT − λ is invertible. . . and of course it is important!
![Page 134: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/134.jpg)
Looking at an object from all directions
We agree that being invertible is an important property of linear operators.
For each λ ∈ k , we have an automorphism T ↦ T − λ of the categoryEndo(FDVect) of operators.
So, given an operator T , it must be important to ask for each λ ∈ k whetherT − λ is invertible. . . and of course it is important!
![Page 135: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/135.jpg)
Looking at an object from all directions
We agree that being invertible is an important property of linear operators.
For each λ ∈ k , we have an automorphism T ↦ T − λ of the categoryEndo(FDVect) of operators.
So, given an operator T , it must be important to ask for each λ ∈ k whetherT − λ is invertible. . .
and of course it is important!
![Page 136: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/136.jpg)
Looking at an object from all directions
We agree that being invertible is an important property of linear operators.
For each λ ∈ k , we have an automorphism T ↦ T − λ of the categoryEndo(FDVect) of operators.
So, given an operator T , it must be important to ask for each λ ∈ k whetherT − λ is invertible. . . and of course it is important!
![Page 137: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/137.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’
: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 138: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/138.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 139: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/139.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 140: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/140.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced
, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 141: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/141.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 142: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/142.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced
, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 143: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/143.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 144: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/144.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced
, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 145: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/145.jpg)
Balanced invariantsLet E be a category. An invariant Φ of objects of E is balanced if it ‘looks atobjects from all directions’: formally, for every automorphism F of E ,
Φ(E1) = Φ(E2) Ô⇒ Φ(F (E1)) = Φ(F (E2))
for E1,E2 ∈ E .
In other words: Φ is balanced if Φ(E) determines Φ(F (E)).
E.g. rings left perfect?Ð→ true,false is not balanced, but
rings (left perf?, right perf?)Ð→ true,false × true,false is.
E.g. compact metric spaces N1Ð→ N is not balanced, but
compact metric spaces (Nr )r>0Ð→ functions R+ → N is.
E.g. linear operators injective?Ð→ true,false is not balanced, but
linear operators eigenvaluesÐ→ subsets of k is.
![Page 146: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/146.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 147: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/147.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 148: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/148.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 149: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/149.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β
,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 150: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/150.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 151: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/151.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 152: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/152.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 153: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/153.jpg)
The (invertible) spectrum is a balanced invariant
We have invariants Spec and Spec× on Endo(FDVect) (the category ofoperators).
This category has some obvious automorphisms: those of the formT ↦ αT + β, where α,β ∈ k with α ≠ 0.
Fix α and β. For all operators T ,
Spec(αT + β) = αSpec(T ) + β,
so Spec(T ) determines Spec(αT + β).
There are also some non-obvious automorphisms of Endo(FDVect)!
Even so, it’s a fact that Spec(T ) determines Spec(Φ(T )) for allautomorphisms Φ of Endo(FDVect).
The same is true of Spec× (for slightly more subtle reasons).
In other words, Spec and Spec× are balanced invariants of linear operators.
![Page 154: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/154.jpg)
5. The theorem
![Page 155: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/155.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes. (Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 156: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/156.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω
such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes. (Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 157: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/157.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes. (Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 158: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/158.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes. (Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 159: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/159.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes.
(Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 160: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/160.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes. (Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 161: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/161.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes. (Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced
, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 162: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/162.jpg)
The theoremTheorem
Spec× is the universal cyclic, balanced invariant of linear operatorson finite-dimensional k-vector spaces.
That is: let Ω be a set and take Φ∶ob (Endo(FDVect))/≅Ð→ Ω such that
i. Φ(g f ) = Φ(f g) whenever Xf //Yg
oo in FDVect;
ii. Φ(T1) = Φ(T2) Ô⇒ Φ(F (T1)) = Φ(F (T2))for all linear operators T1, T2 and automorphisms F of Endo(FDVect).
Then there exists a unique Φ such that
operators Spec× //
Φ++XXXXXXXXXXXXXXXXXXXXXXXXXXXXX finite subsets-with-multiplicity of k×
Φ
Ω
commutes. (Thus, any such Φ is a specialization of Spec×.)
E.g.: tr is cyclic and balanced, and indeed tr(T ) = ∑λ∈k× αT (λ) ⋅ λ.
![Page 163: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/163.jpg)
Imitating the (invertible) spectrum in other categories
What happens if you replace FDVect by a different category?
That is: what if we look for the universal cyclic, balanced invariant ofendomorphisms in C , for some other category C ?
Example: In FinSet, a typical endomorphism looks like this:
The universal cyclic, balanced invariant of endomorphisms in FinSet is
X⤾T ↦ (number of 1-cycles,number of 2-cycles,number of 3-cycles, . . .).
That’s the ‘invertible spectrum’ of an operator on a finite set.
![Page 164: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/164.jpg)
Imitating the (invertible) spectrum in other categories
What happens if you replace FDVect by a different category?
That is: what if we look for the universal cyclic, balanced invariant ofendomorphisms in C , for some other category C ?
Example: In FinSet, a typical endomorphism looks like this:
The universal cyclic, balanced invariant of endomorphisms in FinSet is
X⤾T ↦ (number of 1-cycles,number of 2-cycles,number of 3-cycles, . . .).
That’s the ‘invertible spectrum’ of an operator on a finite set.
![Page 165: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/165.jpg)
Imitating the (invertible) spectrum in other categories
What happens if you replace FDVect by a different category?
That is: what if we look for the universal cyclic, balanced invariant ofendomorphisms in C , for some other category C ?
Example: In FinSet, a typical endomorphism looks like this:
The universal cyclic, balanced invariant of endomorphisms in FinSet is
X⤾T ↦ (number of 1-cycles,number of 2-cycles,number of 3-cycles, . . .).
That’s the ‘invertible spectrum’ of an operator on a finite set.
![Page 166: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/166.jpg)
Imitating the (invertible) spectrum in other categories
What happens if you replace FDVect by a different category?
That is: what if we look for the universal cyclic, balanced invariant ofendomorphisms in C , for some other category C ?
Example: In FinSet, a typical endomorphism looks like this:
The universal cyclic, balanced invariant of endomorphisms in FinSet is
X⤾T ↦ (number of 1-cycles,number of 2-cycles,number of 3-cycles, . . .).
That’s the ‘invertible spectrum’ of an operator on a finite set.
![Page 167: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/167.jpg)
Imitating the (invertible) spectrum in other categories
What happens if you replace FDVect by a different category?
That is: what if we look for the universal cyclic, balanced invariant ofendomorphisms in C , for some other category C ?
Example: In FinSet, a typical endomorphism looks like this:
The universal cyclic, balanced invariant of endomorphisms in FinSet is
X⤾T ↦ (number of 1-cycles,number of 2-cycles,number of 3-cycles, . . .).
That’s the ‘invertible spectrum’ of an operator on a finite set.
![Page 168: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/168.jpg)
Imitating the (invertible) spectrum in other categories
What happens if you replace FDVect by a different category?
That is: what if we look for the universal cyclic, balanced invariant ofendomorphisms in C , for some other category C ?
Example: In FinSet, a typical endomorphism looks like this:
The universal cyclic, balanced invariant of endomorphisms in FinSet is
X⤾T ↦ (number of 1-cycles,number of 2-cycles,number of 3-cycles, . . .).
That’s the ‘invertible spectrum’ of an operator on a finite set.
![Page 169: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/169.jpg)
Imitating the (invertible) spectrum in other categories
What happens if you replace FDVect by a different category?
That is: what if we look for the universal cyclic, balanced invariant ofendomorphisms in C , for some other category C ?
Example: In FinSet, a typical endomorphism looks like this:
The universal cyclic, balanced invariant of endomorphisms in FinSet is
X⤾T ↦ (number of 1-cycles,number of 2-cycles,number of 3-cycles, . . .).
That’s the ‘invertible spectrum’ of an operator on a finite set.
![Page 170: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/170.jpg)
Postscript: Commutative ringsand topos theory
![Page 171: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/171.jpg)
From commutative rings to linear operators
The spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 172: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/172.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R
, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 173: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/173.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 174: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/174.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 175: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/175.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 176: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/176.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 177: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/177.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 178: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/178.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 179: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/179.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 180: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/180.jpg)
From commutative rings to linear operatorsThe spectrum Spec(R) of a (commutative) ring R is the set of prime idealsof R, equipped with:
a certain topology
a certain sheaf of local rings.
The spectrum of a linear operator is a special case:
Given an operator T , put R(T ) = k[x]/(χT (x)).
Then the prime ideals of R(T ) are
(x − λ1), . . . , (x − λm)
where λ1, . . . , λm are the eigenvalues of T .
Moreover, the stalks of the sheaf of local rings have dimensionsαT (λ1), . . . , αT (λm).
Thus, the linear-algebraic spectrum Spec(T ) can be recovered from thering-theoretic spectrum Spec(R(T )).
But how can we understand the ring-theoretic spectrum abstractly?
![Page 181: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/181.jpg)
Hakim’s theorem
Monique Hakim (1986)
Book (1972)
![Page 182: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/182.jpg)
Rings vs. local rings
Fact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint (for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 183: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/183.jpg)
Rings vs. local ringsFact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint (for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 184: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/184.jpg)
Rings vs. local ringsFact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint (for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 185: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/185.jpg)
Rings vs. local ringsFact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint (for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 186: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/186.jpg)
Rings vs. local ringsFact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint (for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 187: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/187.jpg)
Rings vs. local ringsFact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint
(for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 188: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/188.jpg)
Rings vs. local ringsFact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint (for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 189: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/189.jpg)
Rings vs. local ringsFact: The inclusion functor
(local rings) (rings)
has no adjoint.
Idea: Overcome this by allowing the ambient topos to vary.
Let RingTopos be the category of pairs (E ,R) where E is a topos and R isa ring in E .
Let LocRingTopos be the category of pairs (E ,R) where E is a topos andR is a local ring in E .
Fact: The inclusion functor
LocRingTopos RingTopos
has a right adjoint (for completely general reasons).
You can think of the adjoint as constructing the ‘free local ring’ on a ring:but it might live in a different topos from the ring you started with.
![Page 190: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/190.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 191: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/191.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 192: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/192.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 193: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/193.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 194: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/194.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 195: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/195.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 196: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/196.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 197: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/197.jpg)
The spectrum as the free local ring
Let R be a ring.
Then R is a ring in Set, so determines an object (Set,R) of RingTopos.
Also Spec(R) is a topological space, giving a topos Sh(Spec(R)).
It comes with a sheaf of local rings, giving a local ring OR in the toposSh(Spec(R)).
So, the spectrum of R determines an object (Sh(Spec(R)),OR) ofLocRingTopos.
Theorem (Hakim)
The right adjoint to the inclusion LocRingTopos RingTopos maps(Set,R) to (Sh(Spec(R)),OR), for all rings R.
In this sense, Spec is exactly that right adjoint.
![Page 198: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/198.jpg)
Perspectives
Hakim’s theorem describes a universal property of the spectrum of a ring.
The spectrum of a linear operator is a special case of the spectrum of a ring.
So, this gives an abstract characterization of the spectrum of an operator.
However:
To make the step from operators to rings, we used the characteristicpolynomial. What is its place abstractly?
The characterization of the spectrum of an operator coming fromHakim’s theorem is less direct than the one established in this talk,which stays within the topos of sets.
![Page 199: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/199.jpg)
Perspectives
Hakim’s theorem describes a universal property of the spectrum of a ring.
The spectrum of a linear operator is a special case of the spectrum of a ring.
So, this gives an abstract characterization of the spectrum of an operator.
However:
To make the step from operators to rings, we used the characteristicpolynomial. What is its place abstractly?
The characterization of the spectrum of an operator coming fromHakim’s theorem is less direct than the one established in this talk,which stays within the topos of sets.
![Page 200: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/200.jpg)
Perspectives
Hakim’s theorem describes a universal property of the spectrum of a ring.
The spectrum of a linear operator is a special case of the spectrum of a ring.
So, this gives an abstract characterization of the spectrum of an operator.
However:
To make the step from operators to rings, we used the characteristicpolynomial. What is its place abstractly?
The characterization of the spectrum of an operator coming fromHakim’s theorem is less direct than the one established in this talk,which stays within the topos of sets.
![Page 201: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/201.jpg)
Perspectives
Hakim’s theorem describes a universal property of the spectrum of a ring.
The spectrum of a linear operator is a special case of the spectrum of a ring.
So, this gives an abstract characterization of the spectrum of an operator.
However:
To make the step from operators to rings, we used the characteristicpolynomial. What is its place abstractly?
The characterization of the spectrum of an operator coming fromHakim’s theorem is less direct than the one established in this talk,which stays within the topos of sets.
![Page 202: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/202.jpg)
Perspectives
Hakim’s theorem describes a universal property of the spectrum of a ring.
The spectrum of a linear operator is a special case of the spectrum of a ring.
So, this gives an abstract characterization of the spectrum of an operator.
However:
To make the step from operators to rings, we used the characteristicpolynomial. What is its place abstractly?
The characterization of the spectrum of an operator coming fromHakim’s theorem is less direct than the one established in this talk,which stays within the topos of sets.
![Page 203: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/203.jpg)
Perspectives
Hakim’s theorem describes a universal property of the spectrum of a ring.
The spectrum of a linear operator is a special case of the spectrum of a ring.
So, this gives an abstract characterization of the spectrum of an operator.
However:
To make the step from operators to rings, we used the characteristicpolynomial. What is its place abstractly?
The characterization of the spectrum of an operator coming fromHakim’s theorem is less direct than the one established in this talk,which stays within the topos of sets.
![Page 204: Tom Leinster University of Edinburghqpl2016.cis.strath.ac.uk/pdfs/1Leinster.pdfLinear Algebra Done Right 2.Invariants 3.Cyclic invariants 4.Balanced invariants 5.The theorem Outline](https://reader036.vdocument.in/reader036/viewer/2022062508/5febc8b289940e19372c5a23/html5/thumbnails/204.jpg)
Perspectives
Hakim’s theorem describes a universal property of the spectrum of a ring.
The spectrum of a linear operator is a special case of the spectrum of a ring.
So, this gives an abstract characterization of the spectrum of an operator.
However:
To make the step from operators to rings, we used the characteristicpolynomial. What is its place abstractly?
The characterization of the spectrum of an operator coming fromHakim’s theorem is less direct than the one established in this talk,which stays within the topos of sets.