实对称正定矩阵存在平方根的证明

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VIP免费原创最新推荐文章于 2026-06-27 14:28:51 发布 9.7k 阅读 AI 写同款· GEO 优化 ›

矩阵开方 / 矩阵平方根的计算方法及程序实现转载自： https://blog.youkuaiyun.com/DoctorCuiLab/article/details/83048600 阅读详情

$∵A是实对称矩阵,∴存在正交阵P,使P−1AP=Λ=(λ1    λ2    ⋱    λn)⇒A=PΛP−1∵A正定,∴λ1,λ2,...,λn全是正数,令Λ1=(λ1    λ2    ⋱    λn),B=PΛ1P−1则B正定,且有B2=BB=PΛ1P−1PΛ1P−1=PΛ12P−1=PΛP−1=A以上内容编辑：尹蓓\because A是实对称矩阵,\therefore 存在正交阵P,使\\ P^{-1}AP=\Lambda=\left( \begin{array}{cccc} \lambda_1&\, &\, &\,\\ \, & \lambda_2&\, &\, \\ \, &\,&\ddots &\, \\ \, &\,& \, &\lambda_n \\ \end{array} \right)\Rightarrow A= P\Lambda P^{-1}\\ \because A正定,\therefore \lambda_1,\lambda_2,...,\lambda_n全是正数,令\\ \Lambda_1 =\left( \begin{array}{cccc} \sqrt{\lambda_1}&\, &\, &\,\\ \, & \sqrt{\lambda_2}&\, &\, \\ \, &\,&\ddots &\, \\ \, &\,& \, &\sqrt{\lambda_n}\\ \end{array} \right),B=P\Lambda_1P^{-1}\\ 则B正定,且有B^2=BB=P\Lambda_1P^{-1}P\Lambda_1P^{-1}\\=P\Lambda_1^2 P^{-1}=P\Lambda P^{-1}=A\\ 以上内容编辑：尹蓓$