正交矩阵

作者: ltochange | 来源:发表于2021-05-23 12:27 被阅读0次

什么是正交矩阵

满足公式 $AA^{T} =A^{T} A=E$ 的矩阵就是正交矩阵，那么正交矩阵有什么特性呢？

将A表示由行向量组成的矩阵 $A=\left(\begin{array}{c}\alpha_{1} \\ \alpha_{2} \\ \vdots \\ \alpha_{n}\end{array}\right)$ ，则

$AA^\mathrm{T}=\left(\begin{array}{c}\alpha_{1} \\ \alpha_{2} \\ \vdots \\ \alpha_{n}\end{array}\right)\left(\begin{array}{llll}\alpha_{1}^{\mathrm{T}} & \alpha_{2}^{\mathrm{T}} & \cdots & \alpha_{n}^{\mathrm{T}}\end{array}\right)$

$=\left(\begin{array}{c}\alpha_{1} \alpha_{1}^{{T}},\alpha_{1} \alpha_{2}^{\mathrm{T}},\cdots ,\alpha_{1} \alpha_{n}^{\mathrm{T}}\\ \alpha_{2} \alpha_{1}^{\mathrm{T}},\alpha_{2} \alpha_{2}^{\mathrm{T}},\cdots ,\alpha_{2} \alpha_{n}^{\mathrm{T}} \\ \vdots \\ \alpha_{n} \alpha_{1}^{\mathrm{T}},\alpha_{n} \alpha_{2}^{\mathrm{T}},\cdots ,\alpha_{n} \alpha_{n}^{\mathrm{T}}\end{array}\right)= \left(\begin{array}{c}1,0,\cdots ,0\\ 0,1,\cdots ,0 \\ \vdots \\ 0,0,\cdots ,1\end{array}\right)$