每週問題 March 28, 2016

判定兩個冪零矩陣相似的充要條件。

Let A and B be n\times n nonzero matrices.
(a) If A^2=B^2=0, is it true that A and B are similar if and only if \hbox{rank}A=\hbox{rank}B?
(b) If A^3=B^3=0, is it true that A and B are similar if and only if \hbox{rank}A=\hbox{rank}B?

 
參考解答:

A 相似於 B,則 \hbox{rank}A=\hbox{rank}B。底下考慮反向陳述。

(a) 正確。若 A^2=B^2=0,則 AB 是冪零矩陣 (nilpotent matrix),其特徵值皆為 0AB 的最大 Jordan 分塊為 \begin{bmatrix}0&1\\0&0\end{bmatrix}。若 \hbox{rank}A=\hbox{rank}B=r>0,則 AB 都有 r 個 Jordan 分塊 \begin{bmatrix}0&1\\0&0\end{bmatrix},換句話說,AB 有相同的 Jordan 形式,證明 A 相似於 B

(b) 錯誤。若 A^3=B^3=0,則 AB 是冪零矩陣,其特徵值皆為 0AB 的最大 Jordan 分塊為 \begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}。但 \hbox{rank}A=\hbox{rank}B 並不能保證 AB 有相同的 Jordan 形式,例如下列兩個矩陣不相似:

A=\begin{bmatrix} 0&1&0&\vline&0\\ 0&0&1&\vline&0\\ 0&0&0&\vline&0\\ \hline 0&0&0&\vline &0\end{bmatrix},~~B=\begin{bmatrix} 0&1&\vline&0&0\\ 0&0&\vline&0&0\\\hline 0&0&\vline&0&1\\ 0&0&\vline&0&0 \end{bmatrix}

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