logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable HomogeneousSystems of Linear Differential Equations
with Constant Coefficients
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.
4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x)
= ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x)
= ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x
= Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′
= (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.
5. Conversely, every solution of~y′ = A~y can be obtained asabove.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
1. These systems are typically written in matrix form as~y′ = A~y, where A is an n×n matrix and~y is a columnvector with n rows.
2. The theory guarantees that there will always be a set of nlinearly independent solutions {~y1, . . . ,~yn}.
3. Every solution is of the form~y = c1~y1 + · · ·+ cn~yn.4. If A = ΦDΦ−1 and~x solves~x′ = D~x, then
A(Φ~x) = ΦDΦ−1(Φ~x) = ΦD~x = Φ~x′ = (Φ~x)′,
that is,~y = Φ~x solves~y′ = A~y.5. Conversely, every solution of~y′ = A~y can be obtained as
above.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
6. So if we can find a representation A = ΦDΦ−1 so that~x′ = D~x is easy to solve, then~y′ = A~y is also easy to solve.
7. Not every matrix is diagonalizable.8. But if λj is an eigenvalue and~v is a corresponding
eigenvector, then~y = eλjt~v solves~y′ = A~y.9. The multiplicity of the eigenvalue λj is the largest k so that
(λ −λj)k divides the characteristic polynomialp(λ ) = det(A−λ I).
10. If the number of linearly independent eigenvectors for λj isless than the multiplicity, then the matrix is notdiagonalizable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
6. So if we can find a representation A = ΦDΦ−1 so that~x′ = D~x is easy to solve, then~y′ = A~y is also easy to solve.
7. Not every matrix is diagonalizable.8. But if λj is an eigenvalue and~v is a corresponding
eigenvector, then~y = eλjt~v solves~y′ = A~y.9. The multiplicity of the eigenvalue λj is the largest k so that
(λ −λj)k divides the characteristic polynomialp(λ ) = det(A−λ I).
10. If the number of linearly independent eigenvectors for λj isless than the multiplicity, then the matrix is notdiagonalizable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
6. So if we can find a representation A = ΦDΦ−1 so that~x′ = D~x is easy to solve, then~y′ = A~y is also easy to solve.
7. Not every matrix is diagonalizable.
8. But if λj is an eigenvalue and~v is a correspondingeigenvector, then~y = eλjt~v solves~y′ = A~y.
9. The multiplicity of the eigenvalue λj is the largest k so that(λ −λj)k divides the characteristic polynomialp(λ ) = det(A−λ I).
10. If the number of linearly independent eigenvectors for λj isless than the multiplicity, then the matrix is notdiagonalizable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
6. So if we can find a representation A = ΦDΦ−1 so that~x′ = D~x is easy to solve, then~y′ = A~y is also easy to solve.
7. Not every matrix is diagonalizable.8. But if λj is an eigenvalue and~v is a corresponding
eigenvector, then~y = eλjt~v solves~y′ = A~y.
9. The multiplicity of the eigenvalue λj is the largest k so that(λ −λj)k divides the characteristic polynomialp(λ ) = det(A−λ I).
10. If the number of linearly independent eigenvectors for λj isless than the multiplicity, then the matrix is notdiagonalizable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
6. So if we can find a representation A = ΦDΦ−1 so that~x′ = D~x is easy to solve, then~y′ = A~y is also easy to solve.
7. Not every matrix is diagonalizable.8. But if λj is an eigenvalue and~v is a corresponding
eigenvector, then~y = eλjt~v solves~y′ = A~y.9. The multiplicity of the eigenvalue λj is the largest k so that
(λ −λj)k divides the characteristic polynomialp(λ ) = det(A−λ I).
10. If the number of linearly independent eigenvectors for λj isless than the multiplicity, then the matrix is notdiagonalizable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
6. So if we can find a representation A = ΦDΦ−1 so that~x′ = D~x is easy to solve, then~y′ = A~y is also easy to solve.
7. Not every matrix is diagonalizable.8. But if λj is an eigenvalue and~v is a corresponding
eigenvector, then~y = eλjt~v solves~y′ = A~y.9. The multiplicity of the eigenvalue λj is the largest k so that
(λ −λj)k divides the characteristic polynomialp(λ ) = det(A−λ I).
10. If the number of linearly independent eigenvectors for λj isless than the multiplicity, then the matrix is notdiagonalizable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
11. If the multiplicity of λ is at least 2, but the associatedeigenspace is one dimensional, then~vteλ t +~weλ t, with~vbeing an eigenvector and ~w satisfying (A−λ I)~w =~v, isanother, linearly independent, solution of~y′ = A~y.
12. If the multiplicity of λ is at least 3, but the associated
eigenspace is one dimensional, then~vt2
2eλ t +~wteλ t +~xeλ t,
with~v being an eigenvector, ~w satisfying (A−λ I)~w =~v,and~x satisfying (A−λ I)~x =~w, is yet another linearlyindependent solution of~y′ = A~y.
13. There is more, but that’s where matrix exponentials and theJordan Normal Form make things more bearable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
11. If the multiplicity of λ is at least 2, but the associatedeigenspace is one dimensional, then~vteλ t +~weλ t, with~vbeing an eigenvector and ~w satisfying (A−λ I)~w =~v, isanother, linearly independent, solution of~y′ = A~y.
12. If the multiplicity of λ is at least 3, but the associated
eigenspace is one dimensional, then~vt2
2eλ t +~wteλ t +~xeλ t,
with~v being an eigenvector, ~w satisfying (A−λ I)~w =~v,and~x satisfying (A−λ I)~x =~w, is yet another linearlyindependent solution of~y′ = A~y.
13. There is more, but that’s where matrix exponentials and theJordan Normal Form make things more bearable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
11. If the multiplicity of λ is at least 2, but the associatedeigenspace is one dimensional, then~vteλ t +~weλ t, with~vbeing an eigenvector and ~w satisfying (A−λ I)~w =~v, isanother, linearly independent, solution of~y′ = A~y.
12. If the multiplicity of λ is at least 3, but the associated
eigenspace is one dimensional, then~vt2
2eλ t +~wteλ t +~xeλ t,
with~v being an eigenvector, ~w satisfying (A−λ I)~w =~v,and~x satisfying (A−λ I)~x =~w, is yet another linearlyindependent solution of~y′ = A~y.
13. There is more, but that’s where matrix exponentials and theJordan Normal Form make things more bearable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Non-Diagonalizable Systems of LinearDifferential Equations with Constant Coefficients
11. If the multiplicity of λ is at least 2, but the associatedeigenspace is one dimensional, then~vteλ t +~weλ t, with~vbeing an eigenvector and ~w satisfying (A−λ I)~w =~v, isanother, linearly independent, solution of~y′ = A~y.
12. If the multiplicity of λ is at least 3, but the associated
eigenspace is one dimensional, then~vt2
2eλ t +~wteλ t +~xeλ t,
with~v being an eigenvector, ~w satisfying (A−λ I)~w =~v,and~x satisfying (A−λ I)~x =~w, is yet another linearlyindependent solution of~y′ = A~y.
13. There is more, but that’s where matrix exponentials and theJordan Normal Form make things more bearable.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Solve the System~y′ =(
1 −11 3
)~y
det(
1−λ −11 3−λ
)= (1−λ )(3−λ )− (−1) ·1
= 3−4λ +λ2 +1
= λ2−4λ +4
λ1,2 =−(−4)±
√(−4)2−4 ·1 ·42 ·1
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Solve the System~y′ =(
1 −11 3
)~y
det(
1−λ −11 3−λ
)
= (1−λ )(3−λ )− (−1) ·1
= 3−4λ +λ2 +1
= λ2−4λ +4
λ1,2 =−(−4)±
√(−4)2−4 ·1 ·42 ·1
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Solve the System~y′ =(
1 −11 3
)~y
det(
1−λ −11 3−λ
)= (1−λ )(3−λ )− (−1) ·1
= 3−4λ +λ2 +1
= λ2−4λ +4
λ1,2 =−(−4)±
√(−4)2−4 ·1 ·42 ·1
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Solve the System~y′ =(
1 −11 3
)~y
det(
1−λ −11 3−λ
)= (1−λ )(3−λ )− (−1) ·1
= 3−4λ +λ2 +1
= λ2−4λ +4
λ1,2 =−(−4)±
√(−4)2−4 ·1 ·42 ·1
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Solve the System~y′ =(
1 −11 3
)~y
det(
1−λ −11 3−λ
)= (1−λ )(3−λ )− (−1) ·1
= 3−4λ +λ2 +1
= λ2−4λ +4
λ1,2 =−(−4)±
√(−4)2−4 ·1 ·42 ·1
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Solve the System~y′ =(
1 −11 3
)~y
det(
1−λ −11 3−λ
)= (1−λ )(3−λ )− (−1) ·1
= 3−4λ +λ2 +1
= λ2−4λ +4
λ1,2 =−(−4)±
√(−4)2−4 ·1 ·42 ·1
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Solve the System~y′ =(
1 −11 3
)~y
det(
1−λ −11 3−λ
)= (1−λ )(3−λ )− (−1) ·1
= 3−4λ +λ2 +1
= λ2−4λ +4
λ1,2 =−(−4)±
√(−4)2−4 ·1 ·42 ·1
= 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)
−1v1 − 1v2 = 01v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2
, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1
, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1
,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:
(1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)
=(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)
= 2(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Eigenvector for λ = 2(
1−2 −11 3−2
)(v1v2
)=
(00
)−1v1 − 1v2 = 0
1v1 + 1v2 = 0
v1 =−v2, v2 := 1, v1 =−1,~v =(−11
).
Check:(
1 −11 3
)(−11
)=
(−22
)= 2
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)
−1w1 − 1w2 = −11w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2
, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0
, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1
, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:
(1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)
=(
11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)
= 2(
10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
Finding ~w(
1−2 −11 3−2
)(w1w2
)=
(−11
)−1w1 − 1w2 = −1
1w1 + 1w2 = 1
w1 = 1−w2, w2 := 0, w1 = 1, ~w =(
10
).
Check:(
1 −11 3
)(10
)=
(11
)= 2
(10
)+
(−11
)√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
General Solution of the System~y′ =(
1 −11 3
)~y
~y = c1
(−11
)te2t + c2
(10
)e2t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
General Solution of the System~y′ =(
1 −11 3
)~y
~y =
c1
(−11
)te2t + c2
(10
)e2t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
General Solution of the System~y′ =(
1 −11 3
)~y
~y = c1
(−11
)te2t
+ c2
(10
)e2t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients
logo1
Overview When Diagonalization Fails An Example
General Solution of the System~y′ =(
1 −11 3
)~y
~y = c1
(−11
)te2t + c2
(10
)e2t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients