每週問題 May 25, 2015

這是關於分塊矩陣行列式的計算問題。

Let M=\begin{bmatrix}  A&B\\  C&D  \end{bmatrix}, where A and D are square matrices of order m and n, respectively. Let E be an m\times m matrix and F be an n\times m matrix. Prove the following identities.

(a) \begin{vmatrix}  EA&EB\\  C&D  \end{vmatrix}=(\det E)(\det M).
(b) \begin{vmatrix}  A&B\\  C+FA&D+FB  \end{vmatrix}=\det M.

 
參考解答:

(a) 考慮分塊矩陣乘法

\displaystyle  \begin{bmatrix}  EA&EB\\  C&D  \end{bmatrix}=\begin{bmatrix}  E&0\\  0&I  \end{bmatrix}\begin{bmatrix}  A&B\\  C&D  \end{bmatrix}

使用矩陣乘積的行列式可乘公式以及分塊三角矩陣的行列式性質,

\displaystyle  \begin{vmatrix}  EA&EB\\  C&D  \end{vmatrix}=\begin{vmatrix}  E&0\\  0&I  \end{vmatrix}\cdot\begin{vmatrix}  A&B\\  C&D  \end{vmatrix}=(\det E)(\det I)(\det M)=(\det E)(\det M)

(b) 考慮分塊矩陣乘法

\displaystyle  \begin{bmatrix}  A&B\\  C+FA&D+FB  \end{bmatrix}=\begin{bmatrix}  I&0\\  F&I  \end{bmatrix}\begin{bmatrix}  A&B\\  C&D  \end{bmatrix}

等號兩邊計算行列式可得

\displaystyle  \begin{vmatrix}  A&B\\  C+FA&D+FB  \end{vmatrix}=\begin{vmatrix}  I&0\\  F&I  \end{vmatrix}\cdot\begin{vmatrix}  A&B\\  C&D  \end{vmatrix}=(\det I)(\det I)(\det M)=\det M

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