## Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix

## Problem 630

Consider the matrix $A=\begin{bmatrix}

a & -b\\

b& a

\end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$.

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

**(b)** For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$.

**(c)** Diagonalize the matrix $A$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.