每週問題 May 13, 2013

這是正交投影矩陣的界定問題。

Let P be an n\times n idempotent matrix, i.e., P^2=P. Show that P is Hermitian if and only if the column space of P is orthogonal to the nullspace of P, i.e., C(P)\perp N(P).

 
參考解答:

P 是 Hermitian,P^\ast=P,則 N(P)=C(P^\ast)^\perp=C(P)^\perp,證得 C(P)\perp N(P)。反過來說,假設 P^2=PC(P)\perp N(P)。對於 n 維向量 \mathbf{x}P\mathbf{x}=P^2\mathbf{x}\in C(P),或改寫為 P(\mathbf{x}-P\mathbf{x})=\mathbf{0},可知 (\mathbf{x}-P\mathbf{x})\in N(P)。所以,對於任意 n 維向量 \mathbf{x}\mathbf{y}

0=(P\mathbf{x})^\ast(\mathbf{y}-P\mathbf{y})=\mathbf{x}^\ast P^\ast\mathbf{y}-\mathbf{x}^\ast P^\ast P\mathbf{y}

0=(\mathbf{x}-P\mathbf{x})^\ast(P\mathbf{y})=\mathbf{x}^\ast P\mathbf{y}-\mathbf{x}^\ast P^\ast P\mathbf{y}

比較上面兩式,可得 \mathbf{x}^\ast P^\ast\mathbf{y}=\mathbf{x}^\ast P\mathbf{y},即證明 P^\ast=P

PowSol-May-13-13

This entry was posted in pow 內積空間, 每週問題 and tagged , , . Bookmark the permalink.

4 Responses to 每週問題 May 13, 2013

  1. npes_87184 says:

    http://s0.wp.com/latex.php?latex=N%28P%29%3DC%28P%5E%5Cast%29%5E%5Cperp%3DC%28P%29%5E%5Cperp&bg=ffffff&fg=333333&s=0
    這個很顯然嗎?我是用左包含跟右包含去證明這件事情的。
    然後,http://s0.wp.com/latex.php?latex=0%3D%28P%5Cmathbf%7Bx%7D%29%5E%5Cast%28%5Cmathbf%7By%7D-P%5Cmathbf%7By%7D%29%3D%5Cmathbf%7Bx%7D%5E%5Cast+P%5E%5Cast%5Cmathbf%7By%7D-%5Cmathbf%7Bx%7D%5E%5Cast+P%5E%5Cast+P%5Cmathbf%7By%7D&bg=ffffff&fg=333333&s=0
    跟這個http://s0.wp.com/latex.php?latex=0%3D%28%5Cmathbf%7Bx%7D-P%5Cmathbf%7Bx%7D%29%5E%5Cast%28P%5Cmathbf%7By%7D%29%3D%5Cmathbf%7Bx%7D%5E%5Cast+P%5Cmathbf%7By%7D-%5Cmathbf%7Bx%7D%5E%5Cast+P%5E%5Cast+P%5Cmathbf%7By%7D&bg=ffffff&fg=333333&s=0
    想不是很通。

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