每週問題 February 8, 2016

關於兩個正規矩陣之積為正規矩陣的一個充分條件。

Let A and B be n\times n normal matrices. If AB is normal, show that BA is normal.

 
參考解答:

使用正規矩陣的一個充要條件:P=[p_{ij}] 是一個正規矩陣 (即 P^\ast P=PP^\ast) 等價於

\displaystyle \hbox{trace}(P^\ast P)=\sum_{i=1}^n\sum_{j=1}^n\vert p_{ij}\vert^2=\sum_{i=1}^n\vert\lambda_i(P)\vert^2

其中 \lambda_i(P) 表示 P 的第 i 個特徵值。使用跡數循環不變性,以及給定條件 ABAB 是正規矩陣,

\displaystyle \begin{aligned} \hbox{trace}(A^\ast B^\ast BA)&=\hbox{trace}(B^\ast BAA^\ast)\\ &=\hbox{trace}(BB^\ast A^\ast A)\\ &=\hbox{trace}(B^\ast A^\ast AB)\\ &=\sum_{i=1}^n\vert\lambda_i(AB)\vert^2\\ &=\sum_{i=1}^n\vert\lambda_i(BA)\vert^2.\end{aligned}

最後一個等式係因 ABBA 有相同的特徵值 (包含相重特徵值),證畢。

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