Abstract Linear Algebra ISingular Value Decomposition (SVD)

Linear Algebra. Week 10

Dr. Marco A Roque Sol

03 / 17 / 2020

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section

we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again

a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n

linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations

with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where

the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix

A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A

is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued.

If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek

solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutions

of the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form

x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt ,

then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then

it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that

λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be

an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand

v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v

a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector

of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the

coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case,

λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ

is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex,

we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have

complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues and

eigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors

always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear

in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate.

Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus,

if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν;

v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

In this section we consider again a system of n linear homogeneousfirst order differential equations with constant coefficients

x′ = Ax

where the coefficient matrix A is real-valued. If we seek solutionsof the form x = veλt , then it follows that λ must be an eigenvalueand v a corresponding eigenvector of the coefficient matrix A.

In the case, λ is complex, we have complex eigenvalues andeigenvectors always appear in complex-conjugate. Thus, if wehave that

λ±k = µ± i ν; v±k = a± i b

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate

eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors

of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A,

then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued

solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions,

but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but

taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t =

eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition,

then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)

X2(t) =1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two complex-conjugate eigenvalues and eigenvectors of thematrix A, then

X±(t) = e(µ±i ν)t (a± i b)

are complex-valued solutions, but taking in account that

e(µ±i ν)t = eµt (cos(νt)± i sin(νt))

and the principle of superposition, then we have that

X1(t) =1

2

(X+(t) + X−(t)

)X2(t) =

1

2i

(X+(t)− X−(t)

)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two (real) solutions !!!

X1(t) = eµt (acos(νt)− bsin(νt))

X2(t) = eµt (acos(νt) + bsin(νt))

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two

(real) solutions !!!

X1(t) = eµt (acos(νt)− bsin(νt))

X2(t) = eµt (acos(νt) + bsin(νt))

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two (real) solutions !!!

X1(t) = eµt (acos(νt)− bsin(νt))

X2(t) = eµt (acos(νt) + bsin(νt))

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two (real) solutions !!!

X1(t) = eµt (acos(νt)− bsin(νt))

X2(t) = eµt (acos(νt) + bsin(νt))

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

are two (real) solutions !!!

X1(t) = eµt (acos(νt)− bsin(νt))

X2(t) = eµt (acos(νt) + bsin(νt))

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve

the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

x

Solution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find

the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues

of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix

A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.6

Solve the following ODE

x′ = Ax =

3 1 10 2 10 −1 2

xSolution

Let’s find the eigenvalues of the matrix A

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

|A− λI| =

∣∣∣∣∣∣3− λ 1 1

0 2− λ 10 −1 2− λ

∣∣∣∣∣∣ = 0(3− λ)

∣∣∣∣2− λ 1−1 2− λ∣∣∣∣ =

(3− λ)(λ2 − 4λ+ 5) = 0 =⇒

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

|A− λI| =

∣∣∣∣∣∣3− λ 1 1

0 2− λ 10 −1 2− λ

∣∣∣∣∣∣ = 0(3− λ)

∣∣∣∣2− λ 1−1 2− λ∣∣∣∣ =

(3− λ)(λ2 − 4λ+ 5) = 0 =⇒

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

|A− λI| =

∣∣∣∣∣∣3− λ 1 1

0 2− λ 10 −1 2− λ

∣∣∣∣∣∣ = 0

(3− λ)∣∣∣∣2− λ 1−1 2− λ

∣∣∣∣ =(3− λ)(λ2 − 4λ+ 5) = 0 =⇒

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

|A− λI| =

∣∣∣∣∣∣3− λ 1 1

0 2− λ 10 −1 2− λ

∣∣∣∣∣∣ = 0(3− λ)

∣∣∣∣2− λ 1−1 2− λ∣∣∣∣ =

(3− λ)(λ2 − 4λ+ 5) = 0 =⇒

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

|A− λI| =

∣∣∣∣∣∣3− λ 1 1

0 2− λ 10 −1 2− λ

∣∣∣∣∣∣ = 0(3− λ)

∣∣∣∣2− λ 1−1 2− λ∣∣∣∣ =

(3− λ)(λ2 − 4λ+ 5) = 0 =⇒

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3,

λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2=

2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

0 1 10 −1 10 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=

0 1 10 0 20 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

λ1 = 3, λ2,3 =4±

√16− (4)(5)

2= 2± i

If λ1 = 3, then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=0 1 10 −1 1

0 −1 −1

v1v2v3

=0 1 10 0 2

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and

a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding

eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If

λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i ,

then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(1) =

100

If λ2 = 2 + i , then

(A− λ1I) v =

3− λ 1 10 2− λ 10 −1 2− λ

v1v2v3

=

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)

v1v2v3

=1− i 1 10 −i 1

0 −1 −i

v1v2v3

=1− i 1 10 −i 1

0 0 0

v1v2v3

=1− i 0 1− i0 −i 1

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)

v1v2v3

=

1− i 1 10 −i 10 −1 −i

v1v2v3

=1− i 1 10 −i 1

0 0 0

v1v2v3

=1− i 0 1− i0 −i 1

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)

v1v2v3

=1− i 1 10 −i 1

0 −1 −i

v1v2v3

=

1− i 1 10 −i 10 0 0

v1v2v3

=1− i 0 1− i0 −i 1

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)

v1v2v3

=1− i 1 10 −i 1

0 −1 −i

v1v2v3

=1− i 1 10 −i 1

0 0 0

v1v2v3

=

1− i 0 1− i0 −i 10 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

3− (2 + i) 1 10 2− (2 + i) 10 −1 2− (2 + i)

v1v2v3

=1− i 1 10 −i 1

0 −1 −i

v1v2v3

=1− i 1 10 −i 1

0 0 0

v1v2v3

=1− i 0 1− i0 −i 1

0 0 0

v1v2v3

=00

0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and

a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding

eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+

i

010

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

=

a + ib

The corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ib

The corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding

solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions

of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation

are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ;

x(2) = e2t

10−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)−

010

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) +

010

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

and a corresponding eigenvector is

v(2) =

10−1

+ i01

0

= a + ibThe corresponding solutions of the differential equation are

x(1) =

100

e3t ; x(2) = e2t 10

−1

cos(t)− 01

0

sin(t)

x(3) = e2t

10−1

cos(t) + 01

0

sin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian

of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions

is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =

e3te2te2t(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)=

e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t

6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

The Wronskian of these solutions is

W [x(1), x(2), x(3)](t) =

∣∣∣∣∣∣e3t e2tcos(t) e2tcos(t)0 −e2tsin(t) e2tsin(t)0 −e2tcos(t) −e2tcos(t)

∣∣∣∣∣∣ =

e3te2te2t

∣∣∣∣∣∣1 cos(t) cos(t)0 −sin(t) sin(t)0 −cos(t) −cos(t)

∣∣∣∣∣∣ =e3te2te2t

(sin2(t) + cos2(t)

)= e7t 6= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence,

the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions

x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and

x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3)

form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and

the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution

of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system

is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X =

c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X =

c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t +

c2

10−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)−

010

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) +

010

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Hence, the solutions x(1), x(2) and x(3) form a fundamental set,and the general solution of the system is

X = c1x(1) + c2x

(2) + c3x(3) =⇒

X = c1

100

e3t + c2 10

−1

e2tcos(t)− 01

0

e2tsin(t)+

c3

10−1

e2tcos(t) + 01

0

e2tsin(t)

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=

c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is

the direction field associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field

associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with

the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with the system

x ′1x ′2x ′3

=

3 1 10 2 10 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

X =

x1x2x3

=c1e3t + e2t(c2cos(t) + c3sin(t))0 e2t(−c2sin(t) + c3cos(t))

0 −e2t(c2cos(t) + c3sin(t))

Here is the direction field associated with the system

x ′1x ′2x ′3

=3 1 10 2 1

0 −1 2

x1x2x3

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2− λ)2 + 1 = λ2 + λ+ 5

4= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2− λ)2 + 1 = λ2 + λ+ 5

4= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve

the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2− λ)2 + 1 = λ2 + λ+ 5

4= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2− λ)2 + 1 = λ2 + λ+ 5

4= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2− λ)2 + 1 = λ2 + λ+ 5

4= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2− λ)2 + 1 = λ2 + λ+ 5

4= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2− λ)2 + 1 = λ2 + λ+ 5

4= 0

Dr. Marco A Roque Sol Linear Algebra. Week 10

Abstract Linear Algebra ISingular Value Decomposition (SVD)

Complex EigenvaluesRepeated EigenvaluesDiagonalization

Complex Eigenvalues

Example 9.7

Solve the following ODE

x′ = Ax =

(−1/2 1−1 −1/2

)X

Solution

Let’s find

the eigenvalues of the matrix A

|A− λI| =∣∣∣∣−1/2− λ 1−1 −1/2− λ

∣∣∣∣ = 0(−1/2�