Activity 7.1.3

Each of the following matrices is symmetric so the Spectral Theorem tells us that each is orthogonally diagonalizable. The point of this activity is to find an orthogonal diagonalization for each matrix.

To begin, find a basis for each eigenspace. Use this basis to find an orthogonal basis for each eigenspace and put these bases together to find an orthogonal basis for $\real^m$ consisting of eigenvectors. Use this basis to write an orthogonal diagonalization of the matrix.

1. $\begin{bmatrix} 0 & 2 \\ 2 & 3 \end{bmatrix}$.

2. $\begin{bmatrix} 4 & -2 & 14 \\ -2 & 19 & -16 \\ 14 & -16 & 13 \end{bmatrix}$.

3. $\begin{bmatrix} 5 & 4 & 2 \\ 4 & 5 & 2 \\ 2 & 2 & 2 \end{bmatrix}$.

4. Consider the matrix $A = B^TB$ where $B = \begin{bmatrix} 0 & 1 & 2 \\ 2 & 0 & 1 \end{bmatrix}$. Explain how we know that $A$ is symmetric and then find an orthogonal diagonalization of $A$.