Exercise 1

Suppose that $A = \begin{bmatrix} 4 & 2 \\ 2 & 7 \end{bmatrix}$.

1. Find an orthogonal diagonalization of $A$.

2. Evaluate the quadratic form $q_A\left(\twovec11\right)$.

3. Find the unit vector $\uvec$ for which $q_A(\uvec)$ is as large as possible. What is the value of $q_A(\uvec)$ in this direction?

4. Find the unit vector $\uvec$ for which $q_A(\uvec)$ is as small as possible. What is the value of $q_A(\uvec)$ in this direction?