Chapter 5.3, Problem 5E

In Exercises 5 and 6. the matrix A is factored in the form PDP⁻¹. Use the Diagonalization Theorem to find die eigenvalues of A and a basis for each eigenspace.

5. $\left[\begin{array}{ccc}2& 2& 1\\ 1& 3& 1\\ 1& 2& 2\end{array}\right]=\left[\begin{array}{ccc}1& 1& 2\\ 1& 0& -1\\ 1& -1& 0\end{array}\right]\left[\begin{array}{ccc}5& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1/4& 1/2& 1/4\\ 1/4& 1/2& -3/4\\ 1/4& -1/2& 1/4\end{array}\right]$