每週問題 March 4, 2013

這是關於實正交投影矩陣的界定問題。

Let P be an n\times n real matrix. If P=P^2=P^T, show that P is an orthogonal projection matrix, i.e., P\mathbf{x}=\mathbf{x} for every \mathbf{x}\in C(P) and (\mathbf{y}-P\mathbf{y})\in C(P)^{\perp} for every \mathbf{y}\in\mathbb{R}^n. Note that C(P) denotes the column space of P.

 
參考解答:

對於每一 \mathbf{x}\in C(P),必定存在 \mathbf{z}\in\mathbb{R}^n 使得 \mathbf{x}=P\mathbf{z}。等號兩邊同時左乘 P,可得 P\mathbf{x}=P^2\mathbf{z}=P\mathbf{z}=\mathbf{x}。對於任意 \mathbf{y},\mathbf{z}\in\mathbb{R}^n,計算

\begin{aligned}  (\mathbf{y}-P\mathbf{y})^T(P\mathbf{z})&=((I-P)\mathbf{y})^TP\mathbf{z}  =\mathbf{y}^T(I-P)^TP\mathbf{z}\\  &=\mathbf{y}^T(I-P)P\mathbf{z}=\mathbf{y}^T(P-P^2)\mathbf{z}=0,\end{aligned}

故知 (\mathbf{y}-P\mathbf{y})\in C(P)^{\perp}

PowSol-March-4-13

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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s