565

$\bf命题1:$设$A$为正定阵,$B$为实对称阵,则$A$,$B$可同时合同对角化

证明:由$A$正定知,存在可逆阵$P$,使得\[{P^T}AP =
E\]
由$B$实对称知${P^T}BP$实对称,则存在正交阵$Q$,使得
\[{Q^T}{P^T}BPQ = diag\left(
{{\lambda _1}, \cdots ,{\lambda _n}} \right)\]
取$R =
PQ$,则存在可逆阵$R$,使得
\[{R^T}AR = E,{R^T}BR = diag\left( {{\lambda _1}, \cdots
,{\lambda _n}} \right)\]
即$A$,$B$可同时合同对角化

时间: 2024-10-06 13:27:55

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