How to maximize matrix products

Question

Find a unit vector v1 and a unit vector v2, such that the term:

$$v^T \begin{bmatrix} 6 & -2 \\ -2 & 6 \end{bmatrix}v$$

is minimized and maximized, respectively. What are the minimum and maximum values of the above term when v is a unit vector?

Answer 1

This problem uses a basic fact about quadratic forms, symmetric matrices and their eigenvalues and eigenvectors, but you don't need to know anything about the latter to solve it. Let $v=(\cos\theta, \sin\theta)^T$. Then the problem becomes that of finding the maxima and minima of $$(\cos\theta, \sin\theta)\pmatrix{6&-2\\-2&6}\pmatrix{\cos\theta\\ \sin\theta}=6\cos^2\theta-4\cos\theta\sin\theta+6\sin^2\theta=6-2\sin{2\theta}$$ on the interval $0\le\theta<2\pi$. You should be able to handle that on your own.

Answer 2

In the symmetric case, the answers are given by the unit eigenvectors with the smallest and largest eigenvalue (respectively). This occurs in your case.