Activity 7.2.4

This activity explores the relationship between the eigenvalues of a symmetric matrix and its definiteness.

1. Consider the diagonal matrix $D=\begin{bmatrix} 4 & 0 \\ 0 & 2 \\ \end{bmatrix}$ and write its quadratic form $q_D(\xvec)$ in terms of the components of $\xvec=\twovec{x_1}{x_2}$. How does this help you decide whether $D$ is positive definite or not?

2. Now consider $D=\begin{bmatrix} 4 & 0 \\ 0 & 0 \\ \end{bmatrix}$ and write its quadratic form $q_D(\xvec)$ in terms of $x_1$ and $x_2$. What can you say about the definiteness of $D$?

3. If $D$ is a diagonal matrix, what condition on the diagonal entries guarantee that $D$ is

   1. positive definite?

   2. positive semidefinite?

   3. negative definite?

   4. negative semidefinite?

   5. indefinite?

4. Suppose that $A$ is a symmetric matrix with eigenvalues 4 and 2 so that $A=QDQ^T$ where $D=\begin{bmatrix}4 & 0 \\ 0 & 2 \end{bmatrix}$. If $\yvec = Q^T\xvec$, then we have $q_A(\xvec) = q_D(\yvec)$. Explain why this tells us that $A$ is positive definite.

5. Suppose that $A$ is a symmetric matrix with eigenvalues 4 and 0. What can you say about the definiteness of $A$ in this case?

6. What condition on the eigenvalues of a symmetric matrix $A$ guarantees that $A$ is

   1. positive definite?

   2. positive semidefinite?

   3. negative definite?

   4. negative semidefinite?

   5. indefinite?