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7.1 22

Ruby@NTUIM

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}
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.
Ruby@NTUIM

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
Ruby@NTUIM

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]
Ruby@NTUIM

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.
Ruby@NTUIM

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.
Ruby@NTUIM

作業解題 > 7.1 22

Finding eigenvalues and eigenvectors of A
(4+(1-\lambda)(2+\lambda))b=0
Ruby@NTUIM

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
Ruby@NTUIM

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
Ruby@NTUIM

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
Ruby@NTUIM

作業解題 > 7.1 22

Finding eigenvalues and eigenvectors of A^2
(A^2-\lambda I)\vec v = \vec 0
Ruby@NTUIM

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]
Ruby@NTUIM

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.
Ruby@NTUIM

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
Ruby@NTUIM

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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