## Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix

## Problem 585

Consider the Hermitian matrix

\[A=\begin{bmatrix}

1 & i\\

-i& 1

\end{bmatrix}.\]

**(a)** Find the eigenvalues of $A$.

**(b)** For each eigenvalue of $A$, find the eigenvectors.

**(c)** Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.