每週問題 January 25, 2016

證明兩個冪等 (idempotent) 矩陣有相同秩的一個充分條件。

If A is an n\times n matrix, a vector \mathbf{x}\in\mathbb{C}^n is said to be a fixed point of A if A\mathbf{x}=\mathbf{x}. Let P and Q be n\times n idempotent matrices, i.e., P^2=P and Q^2=Q. If the zero vector is the only fixed point of P+Q, show that \hbox{rank}P=\hbox{rank}Q.

 
參考解答:

M=P+Q-I。若 M\mathbf{x}=(P+Q-I)\mathbf{x}=\mathbf{0},即 (P+Q)\mathbf{x}=\mathbf{x},則 \mathbf{x}=\mathbf{0},說明 M 是一個可逆矩陣。考慮

\begin{aligned} PM&=P(P+Q-I)=P^2+PQ-P=PQ\\ MQ&=(P+Q-I)Q=PQ+Q^2-Q=PQ. \end{aligned}

因此,PM=MQ,或 P=MQM^{-1},證明 \hbox{rank}P=\hbox{rank}Q

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