特征值和特征向量-2

$$\boldsymbol{A}=\left[\begin{array}{ccc}a & -1 & c \\ 5 & b & 3 \\ 1-c & 0 & -a\end{array}\right]$$

$$|A|=-1 \\A^* ~feature ~values:\lambda_1 ~\& ~feature ~matrix:[-1,-1,1]^T\\$$

$$A^* \cdot ξ_0=\lambda_1 \cdot ξ_0$$

$$AA^* \cdot ξ_0=\lambda_1 \cdot Aξ_0 \\\because A^*=\frac{|A|}{A}\\\therefore AA^* \cdot ξ_0=\lambda_1 \cdot Aξ_0 =|A|\cdotξ_0=-ξ_0\\ =\lambda_1\cdot A\cdotξ_0\ \\$$

$$\to \lambda_{1}\left[\begin{array}{ccc}a & -1 & c \\ 5 & b & 3 \\ 1-c & 0 & -a\end{array}\right]\left[\begin{array}{c}-1 \\ -1 \\ 1\end{array}\right]=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right]$$

$$\left\{\begin{array}{l}\lambda_{1}(-a+1+c)=1---1 \\ \lambda_{1}(-5-b+3)=1 ---2\\ \lambda_{1}(-1+c-a)=-1---3\end{array}\right.$$

$$a=c$$

$$\lambda_1=1\\ b=-3$$

$$|A|=-1=\left|\begin{array}{ccc}a & -1 & a \\ 5 & -3 & 3 \\ 1-a & 0 & -a\end{array}\right|\\$$

$$\left|\begin{array}{ccc}a & -1 & a \\ 5 & -3 & 3 \\ 1 & -1 & 0\end{array}\right|$$

$$-3+3a+3a-5a=a-3=-1 \\\therefore a=2$$

$$a=c=2\\ \lambda_1=1\\ b=-3$$

$$\boldsymbol{A} \sim\left[\begin{array}{lll}c \\ &b \\ & &c\\ \end{array}\right]=B$$

$$P^{-1}AP=B$$

$$P^{-1}EP=E$$

$$\\A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]\\ \boldsymbol{B A}=\left[\begin{array}{lll}1 a_{11} & -2a_{12} & 3 a_{13} \\ 1 a_{21} & -2a_{22} & 3 a_{23} \\ 1 a_{31} & -2a_{32} & 3 a_{33}\end{array}\right]=A\cdot\left[\begin{array}{lll}1 \\ & -2 \\ & & 3\end{array}\right]$$

$$\left[\begin{array}{lll}1 \\ & -2 \\ & & 3\end{array}\right]=K$$

$$B=A\cdot K \cdot A^{-1} =(A^{-1})^{-1}KA^{-1}=B\\ \therefore \boldsymbol{B} \sim K$$

$$\boldsymbol{A}=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & y & 0 \\ 3 & x & 1\end{array}\right] \sim\left[\begin{array}{lll}-2 & \\ & 1& \\& &1 \end{array}\right]$$

$$\sum_{i=1}^{3} a_{i i}=2+y+1=\sum_{i=1}^{3} \lambda_{i}=2$$

$$y=-1$$

$$\left[\begin{array}{ccc}-1 & -1 & 0 \\ -1 & 2 & 0 \\ -3 & -x & 0\end{array}\right]=\left[\begin{array}{ccc}-1 & -1 & 0 \\ 0 & 3 & 0 \\ -3 & -x & 0\end{array}\right]$$

$$x=K\cdot3$$