Exercise 6

Consider the quadratic form $$q\left(\threevec{x_1}{x_2}{x_3}\right) = 7x_1^2 + 4x_2^2 + 7x_3^2 - 2x_1x_2 -4x_1x_3-2x_2x_3.$$

1. What can you say about the definiteness of the matrix $A$ that defines the quadratic form?

2. Find a matrix $Q$ so that the change of coordinates $\yvec = Q^T\xvec$ transforms the quadratic form into one that has no cross terms. Write the quadratic form in terms of $\yvec$.

3. What are the maximum and minimum values for $q(\uvec)$ among all unit vectors $\uvec$?