多元统计分析-矩阵复习
2022/9/15 23:18:33
本文主要是介绍多元统计分析-矩阵复习,对大家解决编程问题具有一定的参考价值,需要的程序猿们随着小编来一起学习吧!
矩阵代数
特别性质:
1.
若 \(A_{p\times q}~,~B_{q\times p}\) , 则
\[\left|\boldsymbol{I}_{p}+\boldsymbol{A B}\right|=\left|\boldsymbol{I}_{q}+\boldsymbol{B} \boldsymbol{A}\right| \]证明:
\[\begin{array}{c} \because\left[\begin{array}{cc} \boldsymbol{I}_{p} & \boldsymbol{A} \\ \mathbf{0} & \boldsymbol{I}_{q} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{I}_{p} & -\boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{I}_{q} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{I}_{p}+\boldsymbol{A B} & \mathbf{0} \\ \boldsymbol{B} & \boldsymbol{I}_{q} \end{array}\right] \\ {\left[\begin{array}{cc} \boldsymbol{I}_{p} & \mathbf{0} \\ -\boldsymbol{B} & \boldsymbol{I}_{q} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{I}_{p} & -\boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{I}_{q} \end{array}\right]=\left[\begin{array}{lc} \boldsymbol{I}_{p} & -\boldsymbol{A} \\ \mathbf{0} & \boldsymbol{I}_{q}+\boldsymbol{A B} \end{array}\right]} \end{array}\]所以上述两等式两边各取行列式,有
\[\left|\boldsymbol{I}_{p}+\boldsymbol{A B}\right|=\left|\boldsymbol{I}_{q}+\boldsymbol{B A}\right| \]2.
\(\left|A^{-1}\right|=\left|A\right|^{-1}\)
3.
若 \(\boldsymbol{A}\) 和 \(\boldsymbol{B}\) 均为 \(p\) 阶非退化方阵,则
\[\left[\begin{array}{cc} \boldsymbol{A} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{B} \end{array}\right]^{-1}=\left[\begin{array}{cc} \boldsymbol{A}^{-1} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{B}^{-1} \end{array}\right]\]4.
\(\boldsymbol{A}\) 和 \(\boldsymbol{A}^{\prime}\) 有相同的特征值。
5.
若 \(A_{p \times q}\) , \(B_{q \times p}\) ,则 \(A B\) 和 \(B A\) 有相同的非零特征值。
证明:
\[\begin{array}{l} \because\left[\begin{array}{cc} \boldsymbol{I}_{p} & -\boldsymbol{A} \\ \mathbf{0} & \lambda \boldsymbol{I}_{q} \end{array}\right]\left[\begin{array}{cc} \lambda \boldsymbol{I}_{p} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{I}_{q} \end{array}\right]=\left[\begin{array}{cc} \lambda \boldsymbol{I}_{p}-\boldsymbol{A B} & \mathbf{0} \\ \lambda \boldsymbol{B} & \lambda \boldsymbol{I}_{q} \end{array}\right],\\ \left[\begin{array}{cc} \boldsymbol{I}_{p} & \mathbf{0} \\ -\boldsymbol{B} & \lambda \boldsymbol{I}_{q} \end{array}\right]\left[\begin{array}{cc} \lambda \boldsymbol{I}_{p} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{I}_{q} \end{array}\right]=\left[\begin{array}{cc} \lambda \boldsymbol{I}_{p} & \boldsymbol{A} \\ \mathbf{0} & \lambda \boldsymbol{I}_{q}-\boldsymbol{B} \boldsymbol{A} \end{array}\right] \\ \therefore\left|\begin{array}{cc} \lambda \boldsymbol{I}_{p}-\boldsymbol{A B} & \mathbf{0} \\ \lambda \boldsymbol{B} & \lambda \boldsymbol{I}_{q} \end{array}\right|=\left|\begin{array}{cc} \lambda \boldsymbol{I}_{p} & \boldsymbol{A} \\ \mathbf{0} & \lambda \boldsymbol{I}_{q}-\boldsymbol{B} \boldsymbol{A} \end{array}\right| \text { ,即 } \lambda^{q}\left|\lambda \boldsymbol{I}_{p}-\boldsymbol{A B}\right|=\lambda^{p}\left|\lambda \boldsymbol{I}_{q}-\boldsymbol{B} \boldsymbol{A}\right| \end{array}\]这篇关于多元统计分析-矩阵复习的文章就介绍到这儿,希望我们推荐的文章对大家有所帮助,也希望大家多多支持为之网!
- 2024-05-01为什么公共事业机构会偏爱 TiDB :TiDB 数据库在某省妇幼健康管理系统的应用
- 2024-04-26敏捷开发:想要快速交付就必须舍弃产品质量?
- 2024-04-26静态代码分析的这些好处,我竟然都不知道?
- 2024-04-26你在测试金字塔的哪一层?(下)
- 2024-04-26快刀斩乱麻,DevOps让代码评审也自动起来
- 2024-04-262024年最好用的10款ER图神器!
- 2024-04-2203-为啥大模型LLM还没能完全替代你?
- 2024-04-2101-大语言模型发展
- 2024-04-17基于SpringWeb MultipartFile文件上传、下载功能
- 2024-04-14个人开发者,Spring Boot 项目如何部署