每週問題 August 1, 2016

利用冪等矩陣 (idempotent matrix) 計算分塊上三角矩陣的冪。

For the matrix

A=\begin{bmatrix} 1&0&0&1/3&1/3&1/3\\ 0&1&0&1/3&1/3&1/3\\ 0&0&1&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3 \end{bmatrix},

determine A^{300}.

 
參考解答:

寫出 A=\begin{bmatrix} I&B\\ 0&B \end{bmatrix},其中 B=\frac{1}{3}\begin{bmatrix} 1&1&1\\ 1&1&1\\ 1&1&1 \end{bmatrix} 是一個冪等矩陣,即 B^2=B。上式等號兩邊連續乘以 B 可歸納出 B^k=Bk\ge 1,並推得

A^k=\begin{bmatrix} I&B\\ 0&B \end{bmatrix}^k=\begin{bmatrix} I&B+B^2+\cdots+B^k\\ 0&B^k \end{bmatrix}=\begin{bmatrix} I&kB\\ 0&B \end{bmatrix}

因此,

A^{300}=\begin{bmatrix} 1&0&0&100&100&100\\ 0&1&0&100&100&100\\ 0&0&1&100&100&100\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3\\ 0&0&0&1/3&1/3&1/3 \end{bmatrix}

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