## 每週問題 July 22, 2013

Let $A$ be a $2\times 2$ real matrix. If $A^2=-I$, show that $A$ is diagonalizable.

$J^2=M^{-1}A^2M=-M^{-1}M=-I$

$J^2=(\lambda I+N)^2=\lambda^2I^2+2\lambda IN+N^2=\lambda^2I+2\lambda N$

$J=\begin{bmatrix} \pm i&0\\ 0&\pm i \end{bmatrix}$