Exercise 3

Consider the matrix $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$.

1. Find a singular value decomposition of $A$ and verify that it is also an orthogonal diagonalization of $A$.

2. If $A$ is a symmetric, positive semidefinite matrix, explain why a singular value decomposition of $A$ is an orthogonal diagonalization of $A$.