## Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$

## Problem 533

Consider the complex matrix

\[A=\begin{bmatrix}

\sqrt{2}\cos x & i \sin x & 0 \\

i \sin x &0 &-i \sin x \\

0 & -i \sin x & -\sqrt{2} \cos x

\end{bmatrix},\]
where $x$ is a real number between $0$ and $2\pi$.

Determine for which values of $x$ the matrix $A$ is diagonalizable.

When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.