特征值和特征向量-3

Author : 张一极
1024,节日快乐~

一

矩阵可以相似对角化的条件:

定义:A可相似对角化

P^{-1}AP=\Lambda

\boldsymbol{P}=\left[\boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n}\right], \quad \boldsymbol{\Lambda}=\left[\begin{array}{llll}\lambda_{1} & & & \\ & \lambda_{2} & & \\ & & \ddots & \\ & & & \lambda_{n}\end{array}\right]

左乘P:

AP=P\Lambda

代入上式:

\left[A \xi_{1}, A \xi_{2}, \cdots, A \xi_{n}\right]=\left[\lambda_{1} \xi_{1}, \lambda_{2} \xi_{2}, \cdots, \lambda_{n} \xi_{n}\right]

对号入座:

Aξ_{1,2,3,4,5...n}=\lambda_{1,2...n}ξ_{1,2...n}

P可逆,故行列式不为0,(满秩矩阵),即由n个线性无关的向量组成(一行(列)为0,行列式为0)

根据特征向量和特征值的定义可知,A具有n个线性无关的特征向量

通过A有n个不同特征值,亦可知A可相似对角化

Reference From:https://www.bangxuetang.com/points/detail/?sid=301&bid=32&cid=1290&nid=1293&pg=2

$ξ_1=[1,1,1]^T, ξ_2=[2,2,1]^T$ ,是1,1特征值对应的特征向量:

$[x_1,x_2,x_3]$ ,两线性无关向量,乘积为0:

\left\{\begin{array}{l}\left(\xi_{1}, \xi_{3}\right)=x_{1}+x_{2}+x_{3}=0 \\ \left(\xi_{2}, \xi_{3}\right)=2 x_{1}+2 x_{2}+x_{3}=0\end{array}\right.

可知:

x_1=-x_2\\ x_3=0

假设:

ξ_3=[1,-1,0]\\

p=[ξ_1,ξ_2,ξ_3]

得:

\because P^{-1}AP=\Lambda \\------- \\ \therefore \boldsymbol{A}=\boldsymbol{P} \boldsymbol{\Lambda} \boldsymbol{P}^{-1}

\left[\begin{array}{ccc}1 & 2 & 1 \\ 1 & 2 & -1 \\ 1 & 1 & 0\end{array}\right]\left[\begin{array}{lll}1 & \\ & 1 \\ & & -1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 1 \\ 1 & 2 & -1 \\ 1 & 1 & 0\end{array}\right]^{-1}=A

$P^{-1}$ :

P_{11}=1;P_{12}=-1|\left[\begin{array}{cc}1&-1 \\1&0\end{array}\right]|=-1\cdots

$P^*$

P^{-1}=\frac{1}{|P|}A^*

最终

=-\frac{1}{2}\left[\begin{array}{ccc}1 & 2 & 1 \\ 1 & 2 & -1 \\ 1 & 1 & 0\end{array}\right]\left[\begin{array}{ccc}1 & & \\ & 1 & \\ & & -1\end{array}\right]\left[\begin{array}{ccc}1 & 1 & -4 \\ -1 & -1 & 2 \\ -1 & 1 & 0\end{array}\right] =\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]

End
20:53-20201024
节日快乐~