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7.1 22
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7.1 22
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作業解題 > 7.1 22
7.1 Exercise 22
\text{Let}
A=\left[
\begin{matrix}
1 & 4 \\
1 & -2 \\
\end{matrix}
\right]
\text{and verify Exercise 21.}
\text{Find the eigenvalues and eigenvectors of }A^2
\text{Exercise 21}
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作業解題 > 7.1 22
7.1 Exercise 21
\text{Prove that if }\lambda\text{ is an eigenvalue of a matrix }A
\text{with associated eigenvector }\mathbf x\text{, and }k\text{ is a}
\text{positive integer, then }\lambda^k\text{ is an eigenvalue of}
\text{the matrix}
A^K=A\cdot A\cdot \dots\cdot A\text{ (}k\text{ factors)}
\text{with associated eigenvector }\mathbf x.
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
A=\left[
\begin{matrix}
1 & 4 \\
1 & -2 \\
\end{matrix}
\right]
\text{Let }
\vec v=
\left[
\begin{matrix}
a \\
b \\
\end{matrix}
\right]
\text{ where }a, b\in\mathbb R\land\vec v\ne \vec0
\text{be an eigenvector of }A
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\because\vec v\text{ is an eigenvector of }A \\
\therefore A\vec v=\lambda\vec v
Ruby@NTUIM
作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
A\vec v=\lambda\vec v
Ruby@NTUIM
作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\left[
\begin{matrix}
1 & 4 \\
1 & -2 \\
\end{matrix}
\right]
\left[
\begin{matrix}
a \\
b \\
\end{matrix}
\right]=
\lambda
\left[
\begin{matrix}
a \\
b \\
\end{matrix}
\right]
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\left[
\begin{matrix}
a+4b \\
a-2b \\
\end{matrix}
\right]=
\left[
\begin{matrix}
\lambda a \\
\lambda b \\
\end{matrix}
\right]
Ruby@NTUIM
作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\left\{
\begin{matrix}
a+4b & = & \lambda a \\
a-2b & = & \lambda b \\
\end{matrix}
\right.
Ruby@NTUIM
作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\left\{
\begin{matrix}
(1-\lambda)a+4b & = & 0 \\
a-(2+\lambda)b & = & 0 \\
\end{matrix}
\right.
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\left\{
\begin{matrix}
(1-\lambda)a+4b & = & 0 \\
(1-\lambda)a-(1-\lambda)(2+\lambda)b & = & 0 \\
\end{matrix}
\right.
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
(4+(1-\lambda)(2+\lambda))b=0
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
(4+(1-\lambda)(2+\lambda))b=0
\because b\ne0
\therefore 4+(1-\lambda)(2+\lambda)=0
\therefore \lambda^2-\lambda-2=0
\therefore (\lambda-2)(\lambda+1)=0
\therefore \lambda=2\lor\lambda=-1
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\text{When }\lambda=2\text{, }a=4b
(1-\lambda)a+4b = 0
\text{When }\lambda=-1\text{, }a=-2b
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A
\lambda=2
\text{ with associated eigenvector }
\vec v=
\left[
\begin{matrix}
4 \\
1
\end{matrix}
\right]
\lambda=-1
\text{ with associated eigenvector }
\vec v=
\left[
\begin{matrix}
2 \\
-1
\end{matrix}
\right]
\text{Therefore, the matrix }A\text{ has eigenvalues: }
Ruby@NTUIM
作業解題 > 7.1 22
A=\left[
\begin{matrix}
1 & 4 \\
1 & -2 \\
\end{matrix}
\right]
Finding eigenvalues and eigenvectors of A^2
Ruby@NTUIM
作業解題 > 7.1 22
A^2=\left[
\begin{matrix}
1 & 4 \\
1 & -2 \\
\end{matrix}
\right]
\left[
\begin{matrix}
1 & 4 \\
1 & -2 \\
\end{matrix}
\right]
Finding eigenvalues and eigenvectors of A^2
Ruby@NTUIM
作業解題 > 7.1 22
A^2=\left[
\begin{matrix}
5 & -4 \\
-1 & 8 \\
\end{matrix}
\right]
Finding eigenvalues and eigenvectors of A^2
Ruby@NTUIM
作業解題 > 7.1 22
A^2=\left[
\begin{matrix}
5 & -4 \\
-1 & 8 \\
\end{matrix}
\right]
Finding eigenvalues and eigenvectors of A^2
\text{Let }
\vec v=
\left[
\begin{matrix}
a \\
b \\
\end{matrix}
\right]
\text{ where }a, b\in\mathbb R\land\vec v\ne \vec0
\text{be an eigenvector of }A^2
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
(A^2-\lambda I)\vec v = \vec 0
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
\left[
\begin{matrix}
5-\lambda & -4 \\
-1 & 8-\lambda \\
\end{matrix}
\right]
\left[
\begin{matrix}
a \\
b \\
\end{matrix}
\right]=
\left[
\begin{matrix}
0 \\
0 \\
\end{matrix}
\right]
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
\left\{
\begin{matrix}
(5-\lambda)a-4b & = & 0 \\
-a+(8-\lambda)b & = & 0 \\
\end{matrix}
\right.
Ruby@NTUIM
作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
\left\{
\begin{matrix}
(5-\lambda)a-4b & = & 0 \\
(5-\lambda)a+(5-\lambda)(\lambda-8)b & = & 0 \\
\end{matrix}
\right.
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
(-\lambda^2+13\lambda-36)b=0
\because b\ne0
\therefore-\lambda^2+13\lambda-36=0
\therefore\lambda^2-13\lambda+36=0
\therefore (\lambda-4)(\lambda-9)=0
\therefore \lambda=4\lor\lambda=9
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作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
(5-\lambda)a-4b = 0
\text{When }\lambda=4\text{, }a=4b
\text{When }\lambda=9\text{, }a=-b
Ruby@NTUIM
作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
\lambda=4
\text{ with associated eigenvector }
\vec v=
\left[
\begin{matrix}
4 \\
1
\end{matrix}
\right]
\lambda=9
\text{ with associated eigenvector }
\vec v=
\left[
\begin{matrix}
1 \\
-1
\end{matrix}
\right]
\text{Therefore, the matrix }A^2\text{ has eigenvalues: }
Ruby@NTUIM
作業解題 > 7.1 22
Finding eigenvalues and eigenvectors of A^2
\lambda=2
\text{ with associated eigenvector }
\vec v=
\left[
\begin{matrix}
4 \\
1
\end{matrix}
\right]
\text{The matrix }A\text{ has positive eigenvalue}
\lambda=4
\text{ with associated eigenvector }
\vec v=
\left[
\begin{matrix}
4 \\
1
\end{matrix}
\right]
\text{The matrix }A^2\text{ has positive eigenvalue}
Ruby@NTUIM
作業解題 > 7.1 22
7.1 Exercise 21
\text{Prove that if }\lambda\text{ is an eigenvalue of a matrix }A
\text{with associated eigenvector }\mathbf x\text{, and }k\text{ is a}
\text{positive integer, then }\lambda^k\text{ is an eigenvalue of}
\text{the matrix}
A^K=A\cdot A\cdot \dots\cdot A\text{ (}k\text{ factors)}
\text{with associated eigenvector }\mathbf x.
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7.1 22
作業解題
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