Exercise 2

Consider the matrix $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \\ \end{bmatrix}$ whose eigenvalues are $\lambda_1=5$, $\lambda_2=-1$, and $\lambda_3 = -1$.

1. Explain why $A$ is orthogonally diagonalizable.

2. Find an orthonormal basis for the eigenspace $E_5$.

3. Find a basis for the eigenspace $E_{-1}$.

4. Now find an orthonormal basis for $E_{-1}$.

5. Find matrices $D$ and $Q$ such that $A=QDQ^T$.