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 $U^{-1}AU=D$.
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