Exercise 8

Determine whether the following statements are true or false and explain your reasoning.

1. If $A$ is an indefinite matrix, we can't know whether it is positive definite or not.

2. If the smallest eigenvalue of $A$ is 3, then $A$ is positive definite.

3. If $C$ is the covariance matrix associated with a dataset, then $C$ is positive semidefinite.

4. If $A$ is a symmetric $2\times2$ matrix and the maximum and minimum values of $q_A(\uvec)$ occur at $\twovec10$ and $\twovec01$, then $A$ is diagonal.

5. If $A$ is negative definite and $Q$ is an orthogonal matrix with $B = QAQ^T$, then $B$ is negative definite.