## 每週問題 October 13, 2014

Let $A$ and $B$ be $n\times n$ real orthogonal projection matrices, i.e., $A^T=A=A^2$, $B^T=B=B^2$. Show that $C(A)=C(B)$ implies $A=B$. Note that $C(X)$ denotes the column space or range of matrix $X$.

$\displaystyle (A-B)^2\mathbf{x}=A^2\mathbf{x}+B^2\mathbf{x}-AB\mathbf{x}-BA\mathbf{x}=A\mathbf{x}+B\mathbf{x}-A(B\mathbf{x})-B(A\mathbf{x})=\mathbf{0}$