每週問題 August 26, 2013

這是關於複矩陣形成的特殊行列式的計算問題。

Let M=A+iB, where A and B are n\times n real matrices and i=\sqrt{-1}. Show that

\begin{vmatrix}  A&iB\\  iB&A  \end{vmatrix}=\vert\det M\vert^2.

 
參考解答:

使用分塊矩陣乘法和行列式性質,可得

\displaystyle\begin{aligned}  \begin{vmatrix}  A&iB\\  iB&A  \end{vmatrix}&=\begin{vmatrix}  A+iB&iB\\  A+iB&A  \end{vmatrix}\\  &=\begin{vmatrix}  I&iB\\  I&A  \end{vmatrix}\begin{vmatrix}  A+iB&0\\  0&I  \end{vmatrix}\\  &=\begin{vmatrix}  I&iB\\  0&A-iB  \end{vmatrix}\begin{vmatrix}  A+iB&0\\  0&I  \end{vmatrix}\\  &=\det(A-iB)\det(A+iB)\\  &=\overline{\det(A+iB)}\det(A+iB)\\  &=\vert \det M\vert^2.\end{aligned}

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