## 每週問題 March 4, 2013

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 $P$.

\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}

PowSol-March-4-13