每週問題 March 13, 2017

任一正規矩陣 (normal matrix) 可表示為一個正規矩陣的平方。

Let A be a normal matrix, i.e., A^\ast A=AA^\ast. Prove that there exists a normal matrix B such that A=B^2.

 
參考解答:

正規矩陣的一個充要條件是可么正對角化 (unitarily diagonalizable)。令 A 為一個 n\times n 階正規矩陣。寫出么正對角化形式 A=U\Lambda U^\ast,其中 U 是么正矩陣 (unitary matrix),U^\ast=U^{-1}\Lambda=\hbox{diag}(\lambda_1,\ldots,\lambda_n)。令 B=UDU^\ast,其中 D=\hbox{diag}(\sqrt{\lambda_1},\ldots,\sqrt{\lambda_n})。因此,B 是一個正規矩陣 (已具有么正對角化形式) 且

B^2=UDU^\ast UDU^\ast=UD^2U^\ast=U\Lambda U^\ast=A

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