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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

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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

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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