## 每週問題 June 16, 2014

Let $A$ be a real symmetric matrix. If $\mathbf{x}$ and $\mathbf{y}$ are eigenvectors of $A$, corresponding to distinct eigenvalues, show that $\mathbf{x}$ and $\mathbf{y}$ are orthogonal.

$\displaystyle (A\mathbf{x})^T\mathbf{y}=\mathbf{x}^TA^T\mathbf{y}=\mathbf{x}^T(A\mathbf{y})$

$\displaystyle \lambda\mathbf{x}^T\mathbf{y}=\mu\mathbf{x}^T\mathbf{y}$

$\lambda\neq\mu$，證得 $\mathbf{x}^T\mathbf{y}=0$