Activity 7.2.2

Let's look at some more examples of quadratic forms.

1. Consider the symmetric matrix $D = \begin{bmatrix} 3 & 0 \\ 0 & -1 \\ \end{bmatrix}$. Write the quadratic form $q_D(\xvec)$ defined by $D$ in terms of the components of $\xvec=\twovec{x_1}{x_2}$. What is the value of $q_D\left(\twovec2{-4}\right)$?

2. Given the symmetric matrix $A=\begin{bmatrix} 2 & 5 \\ 5 & -3 \end{bmatrix}$, write the quadratic form $q_A(\xvec)$ defined by $A$ and evaluate $q_A\left(\twovec{2}{-1}\right)$.

3. Suppose that $q\left(\twovec{x_1}{x_2}\right) = 3x_1^2 - 4x_1x_2 + 4x_2^2$. Find a symmetric matrix $A$ such that $q$ is the quadratic form defined by $A$.

4. Suppose that $q$ is a quadratic form and that $q(\xvec) = 3$. What is $q(2\xvec)$? $q(-\xvec)$? $q(10\xvec)$?

5. Suppose that $A$ is a symmetric matrix and $q_A(\xvec)$ is the quadratic form defined by $A$. Suppose that $\xvec$ is an eigenvector of $A$ with associated eigenvalue -4 and with length 7. What is $q_A(\xvec)$?