每週問題 June 10, 2013

這是一個分塊矩陣的行列式問題。

Let A, B, C, D be n\times n matrices. Show that if A is nonsingular, then

\begin{vmatrix}  A&B\\  C&D  \end{vmatrix}=(\det A)(\det (D-CA^{-1}B)).

 
參考解答:

A 可逆,考慮下列分塊乘法,

\begin{bmatrix}  I&0\\  -CA^{-1}&I  \end{bmatrix}\begin{bmatrix}  A&B\\  C&D  \end{bmatrix}=\begin{bmatrix}  A&B\\  0&D-CA^{-1}B  \end{bmatrix}

等號兩邊取行列式,

\begin{vmatrix}  I&0\\  -CA^{-1}&I  \end{vmatrix}\cdot\begin{vmatrix}  A&B\\  C&D  \end{vmatrix}=\begin{vmatrix}  A&B\\  0&D-CA^{-1}B  \end{vmatrix}

因為分塊三角矩陣的行列式等於主對角分塊行列式之積,推得

\begin{vmatrix}  A&B\\  C&D  \end{vmatrix}=\begin{vmatrix}  A  \end{vmatrix}\cdot\begin{vmatrix}  D-CA^{-1}B  \end{vmatrix}

PowSol-June-10-13

This entry was posted in pow 行列式, 每週問題 and tagged , . Bookmark the permalink.

發表迴響

在下方填入你的資料或按右方圖示以社群網站登入:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / 變更 )

Twitter picture

You are commenting using your Twitter account. Log Out / 變更 )

Facebook照片

You are commenting using your Facebook account. Log Out / 變更 )

Google+ photo

You are commenting using your Google+ account. Log Out / 變更 )

連結到 %s