inverse eigenvalue problems
TRANSCRIPT
Introduction One Simple Algorithm Applications
Inverse Eigenvalue ProblemsConstructing Matrices with Prescribed Eigenvalues
N. Jackson
Department of MathematicsCollege of the Redwoods
Math 45 Term Project, Fall 2010
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Outline
IntroductionEigenvalues and EigenvectorsInverse Eigenvalue Problems (IEP’s)
One Simple AlgorithmHeuvers’ AlgorithmProofAn ExampleBenefits and Drawbacks
Applications
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Eigenvalues and Eigenvectors
What are Eigenvalues and Eigenvectors?
I An eigenvalue is “any number such that a given squarematrix minus that number times the identity matrix has azero determinant” [2].
I Ax = λx
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Eigenvalues and Eigenvectors
What are Eigenvalues and Eigenvectors?
I An eigenvalue is “any number such that a given squarematrix minus that number times the identity matrix has azero determinant” [2].
I Ax = λx
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Inverse Eigenvalue Problems (IEP’s)
Inverse Eigenvalue Problems (IEP’s)
I A well-studied yet continually developing branch of LinearAlgebra concerning construction of matrices from spectraldata.[3]
I Two basic components: solvability and computability.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Inverse Eigenvalue Problems (IEP’s)
Inverse Eigenvalue Problems (IEP’s)
I A well-studied yet continually developing branch of LinearAlgebra concerning construction of matrices from spectraldata.[3]
I Two basic components: solvability and computability.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Heuvers’ Algorithm
Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors
I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.
I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.
I Define µj =√λj − τ and bj = µjpj , and let B be the matrix
comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with
the above eigenvectors and eigenvalues.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Heuvers’ Algorithm
Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors
I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.
I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.
I Define µj =√λj − τ and bj = µjpj , and let B be the matrix
comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with
the above eigenvectors and eigenvalues.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Heuvers’ Algorithm
Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors
I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.
I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.
I Define µj =√λj − τ and bj = µjpj , and let B be the matrix
comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with
the above eigenvectors and eigenvalues.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Heuvers’ Algorithm
Konrad Heuvers’ AlgorithmSymmetric Matrices with Prescribed Eigenvalues and Eigenvectors
I Let {p1,p2, . . . ,pn} be an arbitrary orthonormal basis forRn. These will become the eigenvectors.
I Let λ1, λ2, ..., λn be n arbitrary real numbers (the desiredeigenvalues) and τ be any real number such that τ ≤ λj forj = 1,2, ...,n.
I Define µj =√λj − τ and bj = µjpj , and let B be the matrix
comprised of the column vectors b1,b2, ...bn.I Let S be the matrix S = BBT + τ I, a symmetric matrix with
the above eigenvectors and eigenvalues.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof of Heuvers’ Algorithm
I The columns of B are(b1,b2, . . . ,bn) = (µ1p1, µ2p2, . . . , µnpn).
I The rows of BT are of the form µipTi .
I It must be shown that Spj = λjpj .
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof of Heuvers’ Algorithm
I The columns of B are(b1,b2, . . . ,bn) = (µ1p1, µ2p2, . . . , µnpn).
I The rows of BT are of the form µipTi .
I It must be shown that Spj = λjpj .
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof of Heuvers’ Algorithm
I The columns of B are(b1,b2, . . . ,bn) = (µ1p1, µ2p2, . . . , µnpn).
I The rows of BT are of the form µipTi .
I It must be shown that Spj = λjpj .
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof (cont., 2)
Spj = (BBT + τ I)pj
= BBT pj + τpj
= [µ1p1, µ2p2, . . . , µnpn]
µ1pT
1µ2pT
2...
µnpTn
pj + τpj
= [µ1p1, µ2p2, . . . , µnpn]
µ1pT
1 pjµ2pT
2 pj...
µnpTn pj
+ τpj
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof (cont., 3)
I Column vector all zeros except pj dotted with itself is one.I
. . . = [µ1p1, µ2p2, . . . , µnpn]
µ1pT
1 pjµ2pT
2 pj...
µnpTn pj
+ τpj
= [µ1p1, µ2p2, . . . , µnpn]
0...µj...0
+ τpj
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof (cont., 3)
I Column vector all zeros except pj dotted with itself is one.I
. . . = [µ1p1, µ2p2, . . . , µnpn]
µ1pT
1 pjµ2pT
2 pj...
µnpTn pj
+ τpj
= [µ1p1, µ2p2, . . . , µnpn]
0...µj...0
+ τpj
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof (cont., 4)
I
= µ2j pj + τpj
= (µ2j + τ)pj
= λjpj
I We have shown that Spj = λjpj , therefore each vector pjand corresponding scalar λj are an eigenvector andeigenvalue for the matrix S.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Proof
Proof (cont., 4)
I
= µ2j pj + τpj
= (µ2j + τ)pj
= λjpj
I We have shown that Spj = λjpj , therefore each vector pjand corresponding scalar λj are an eigenvector andeigenvalue for the matrix S.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example
I Arbitrary orthonormal basis for R2:
BR2 =
{[√2/2√
2/2
],
[−√
2/2√
2/2
]}
I For simplicity of computing µ1 and µ2, we’ll choose λ1 = 2,λ2 = 5, and τ = 1.
I
µ1 =√λ1 − τ =
√2− 1 = 1
µ2 =√λ2 − τ =
√5− 1 = 2
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example
I Arbitrary orthonormal basis for R2:
BR2 =
{[√2/2√
2/2
],
[−√
2/2√
2/2
]}
I For simplicity of computing µ1 and µ2, we’ll choose λ1 = 2,λ2 = 5, and τ = 1.
I
µ1 =√λ1 − τ =
√2− 1 = 1
µ2 =√λ2 − τ =
√5− 1 = 2
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example
I Arbitrary orthonormal basis for R2:
BR2 =
{[√2/2√
2/2
],
[−√
2/2√
2/2
]}
I For simplicity of computing µ1 and µ2, we’ll choose λ1 = 2,λ2 = 5, and τ = 1.
I
µ1 =√λ1 − τ =
√2− 1 = 1
µ2 =√λ2 − τ =
√5− 1 = 2
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 2)
I Create the matrix B, composed of the columns b1 and b2:I
B = [b1,b2]
= [µ1p1, µ2p2]
=
[1
[√2/2√
2/2
],2
[−√
2/2√
2/2
]]
=
[√2/2 −
√2
√2/2
√2
]
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 2)
I Create the matrix B, composed of the columns b1 and b2:I
B = [b1,b2]
= [µ1p1, µ2p2]
=
[1
[√2/2√
2/2
],2
[−√
2/2√
2/2
]]
=
[√2/2 −
√2
√2/2
√2
]
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 3)
Now we can create our matrix S:
S = BBT + τ I
=
[√2/2 −
√2
√2/2
√2
][√2/2
√2/2
−√
2√
2
]+ 1
[1 00 1
]
=
[5/2 −3/2
−3/2 5/2
]+
[1 00 1
]
=
[7/2 −3/2
3/2 7/2
]
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 4)
Check this solution by first finding the eigenvalues of S:
det(S − λI) = 0∣∣∣∣∣7/2− λ −3/2
−3/2 7/2− λ
∣∣∣∣∣ = 0
(7/2− λ)2 − (−3/2)2 = 0
49/4− 7λ+ λ2 − 9/4 = 0
λ2 − 7λ+ 10 = 0(λ− 2)(λ− 5) = 0
λ = 2,5
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 5)
...and then by finding the corresponding eigenvectors:I
(S − 2I)x = 0[3/2 −3/2−3/2 3/2
] [xy
]=
[00
][xy
]=
[11
]I
(S − 5I)x = 0[−3/2 −3/2−3/2 −3/2
] [xy
]=
[00
][xy
]=
[−11
]Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 5)
...and then by finding the corresponding eigenvectors:I
(S − 2I)x = 0[3/2 −3/2−3/2 3/2
] [xy
]=
[00
][xy
]=
[11
]I
(S − 5I)x = 0[−3/2 −3/2−3/2 −3/2
] [xy
]=
[00
][xy
]=
[−11
]Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 6)
I Note that the eigenvectors of the S we created with thealgorithm are scalar multiples of the eigenvectors wechose beforehand.
I Since the nullspaces of S − 2I and S − 5I are both closedunder scalar multiplication, the eigenvectors we foundconfirm the validity of the algorithm.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 6)
I Note that the eigenvectors of the S we created with thealgorithm are scalar multiples of the eigenvectors wechose beforehand.
I Since the nullspaces of S − 2I and S − 5I are both closedunder scalar multiplication, the eigenvectors we foundconfirm the validity of the algorithm.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
An Example
An Example (cont., 7)
x
y [11
][−11
][√
2/2√2/2
][−√
2/2√2/2
]
Figure: Eigenvectors of matrix S.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Benefits and Drawbacks
Benefits and Drawbacksof Heuvers’ Algorithm
I Simple to understand and compute.I Always creates symmetric matrices, must normalize
eigenvectors first.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Benefits and Drawbacks
Benefits and Drawbacksof Heuvers’ Algorithm
I Simple to understand and compute.I Always creates symmetric matrices, must normalize
eigenvectors first.
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Applicationsof Inverse Eigenvalue Problems
I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].
I Particle physicsI Molecular spectroscopyI Geophysics
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Applicationsof Inverse Eigenvalue Problems
I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].
I Particle physicsI Molecular spectroscopyI Geophysics
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Applicationsof Inverse Eigenvalue Problems
I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].
I Particle physicsI Molecular spectroscopyI Geophysics
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
Applicationsof Inverse Eigenvalue Problems
I Found in applications where goal is finding physicalparameters of a system based on known behavior orconstructing a system with physical parameters resulting ina desired dynamical behavior [3].
I Particle physicsI Molecular spectroscopyI Geophysics
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
For Further Reading I
G. Strang.Introduction to Linear Algebra, Fourth Edition.Wesley-Cambridge, 2009.
Trustees of Princeton UniversityWordNet A Lexical Database for English, 2010http://wordnetweb.princeton.edu/perl/webwn?s=eigenvalue
Inverse Eigenvalue Problems College of the Redwoods
Introduction One Simple Algorithm Applications
For Further Reading II
M. Chu, G. Golub.Inverse Eigenvalue Problems: Theory and Applications.Department of Mathematics, North Carolina StateUniversity, 2001http://www4.ncsu.edu/~mtchu/Research/Lectures/Iep/preface.ps
K. Heuvers.Symmetric Matrices with Prescribed Eigenvalues andEigenvectorsMathematics Magazine, Vol. 55, No. 2. (Mar., 1982), pp.106-111
Inverse Eigenvalue Problems College of the Redwoods