## A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix

## Problem 584

Prove that the matrix

\[A=\begin{bmatrix}

0 & 1\\

-1& 0

\end{bmatrix}\]
is diagonalizable.

Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix.

That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal matrix.