Non-Diagonalizable Homogeneous Systems of Linear Differential - - PowerPoint PPT Presentation

logo1 Overview When Diagonalization Fails An Example

Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Bernd Schr¨

• der

Bernd Schr¨

• der

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 Linear Differential Equations with Constant Coefficients

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

• y1,...,

yn}.

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly independent solutions {

• y1,...,

yn}.

• 3. Every solution is of the form

y = c1

• y1 +···+cn
• yn.

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly 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)

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly 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)

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly 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

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly 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′

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly 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)′,

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly 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.

Bernd Schr¨

• der

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 Linear Differential 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 column vector with n rows.

• 2. The theory guarantees that there will always be a set of n

linearly 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 Schr¨

• der

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 Linear Differential Equations with Constant Coefficients

Bernd Schr¨

• der

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 Linear Differential 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.

Bernd Schr¨

• der

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 Linear Differential 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.

Bernd Schr¨

• der

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 Linear Differential 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.

Bernd Schr¨

• der

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 Linear Differential 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 polynomial p(λ) = det(A−λI).

Bernd Schr¨

• der

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 Linear Differential 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 polynomial p(λ) = det(A−λI).

• 10. If the number of linearly independent eigenvectors for λj is

less than the multiplicity, then the matrix is not diagonalizable.

Bernd Schr¨

• der

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 Linear Differential Equations with Constant Coefficients

Bernd Schr¨

• der

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 Linear Differential Equations with Constant Coefficients

• 11. If the multiplicity of λ is at least 2, but the associated

eigenspace is one dimensional, then vteλt + weλt, with v being an eigenvector and w satisfying (A−λI) w = v, is another, linearly independent, solution of y′ = A

• y.

Bernd Schr¨

• der

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 Linear Differential Equations with Constant Coefficients

• 11. If the multiplicity of λ is at least 2, but the associated

eigenspace is one dimensional, then vteλt + weλt, with v being an eigenvector and w satisfying (A−λI) w = v, is another, 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 2 eλt + wteλt + xeλt, with v being an eigenvector, w satisfying (A−λI) w = v, and x satisfying (A−λI)

• x =

w, is yet another linearly independent solution of y′ = A

• y.

Bernd Schr¨

• der

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 Linear Differential Equations with Constant Coefficients

• 11. If the multiplicity of λ is at least 2, but the associated

eigenspace is one dimensional, then vteλt + weλt, with v being an eigenvector and w satisfying (A−λI) w = v, is another, 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 2 eλt + wteλt + xeλt, with v being an eigenvector, w satisfying (A−λI) w = v, and x satisfying (A−λI)

• x =

w, is yet another linearly independent solution of y′ = A

• y.
• 13. There is more, but that’s where matrix exponentials and the

Jordan Normal Form make things more bearable.

Bernd Schr¨

• der

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 −1 1 3

• y

Bernd Schr¨

• der

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 −1 1 3

• y

det 1−λ −1 1 3−λ

• Bernd Schr¨
• der

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 −1 1 3

• y

det 1−λ −1 1 3−λ

• =

(1−λ)(3−λ)−(−1)·1

Bernd Schr¨

• der

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 −1 1 3

• y

det 1−λ −1 1 3−λ

• =

(1−λ)(3−λ)−(−1)·1 = 3−4λ +λ 2 +1

Bernd Schr¨

• der

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 −1 1 3

• y

det 1−λ −1 1 3−λ

• =

(1−λ)(3−λ)−(−1)·1 = 3−4λ +λ 2 +1 = λ 2 −4λ +4

Bernd Schr¨

• der

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 −1 1 3

• y

det 1−λ −1 1 3−λ

• =

(1−λ)(3−λ)−(−1)·1 = 3−4λ +λ 2 +1 = λ 2 −4λ +4 λ1,2 = −(−4)±

• (−4)2 −4·1·4

2·1

Bernd Schr¨

• der

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 −1 1 3

• y

det 1−λ −1 1 3−λ

• =

(1−λ)(3−λ)−(−1)·1 = 3−4λ +λ 2 +1 = λ 2 −4λ +4 λ1,2 = −(−4)±

• (−4)2 −4·1·4

2·1 = 2

Bernd Schr¨

• der

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 −1 1 3−2 v1 v2

• =
• Bernd Schr¨
• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 =

Bernd Schr¨

• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2

Bernd Schr¨

• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1

Bernd Schr¨

• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1, v1 = −1

Bernd Schr¨

• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1, v1 = −1, v = −1 1

• .

Bernd Schr¨

• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1, v1 = −1, v = −1 1

• .

Check:

Bernd Schr¨

• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1, v1 = −1, v = −1 1

• .

Check: 1 −1 1 3 −1 1

• Bernd Schr¨
• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1, v1 = −1, v = −1 1

• .

Check: 1 −1 1 3 −1 1

• =

−2 2

• Bernd Schr¨
• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1, v1 = −1, v = −1 1

• .

Check: 1 −1 1 3 −1 1

• =

−2 2

• = 2

−1 1

• Bernd Schr¨
• der

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 −1 1 3−2 v1 v2

• =
• −1v1

− 1v2 = 1v1 + 1v2 = v1 = −v2, v2 := 1, v1 = −1, v = −1 1

• .

Check: 1 −1 1 3 −1 1

• =

−2 2

• = 2

−1 1

• √

Bernd Schr¨

• der

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 −1 1 3−2 w1 w2

• =

−1 1

• Bernd Schr¨
• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1

Bernd Schr¨

• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2

Bernd Schr¨

• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0

Bernd Schr¨

• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0, w1 = 1

Bernd Schr¨

• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0, w1 = 1, w = 1

• .

Bernd Schr¨

• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0, w1 = 1, w = 1

• .

Check:

Bernd Schr¨

• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0, w1 = 1, w = 1

• .

Check: 1 −1 1 3 1

• Bernd Schr¨
• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0, w1 = 1, w = 1

• .

Check: 1 −1 1 3 1

• =

1 1

• Bernd Schr¨
• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0, w1 = 1, w = 1

• .

Check: 1 −1 1 3 1

• =

1 1

• = 2

1

• +

−1 1

• Bernd Schr¨
• der

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 −1 1 3−2 w1 w2

• =

−1 1

• −1w1

− 1w2 = −1 1w1 + 1w2 = 1 w1 = 1−w2, w2 := 0, w1 = 1, w = 1

• .

Check: 1 −1 1 3 1

• =

1 1

• = 2

1

• +

−1 1

• √

Bernd Schr¨

• der

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 −1 1 3

• y

Bernd Schr¨

• der

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 −1 1 3

• y
• y =

Bernd Schr¨

• der

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 −1 1 3

• y
• y = c1

−1 1

• te2t

Bernd Schr¨

• der

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 −1 1 3

• y
• y = c1

−1 1

• te2t +c2

1

• e2t

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients