I have a host of questions that require me to maximise vectors/matrices. This is from an economic module so much of this material is new to me. Anyway there is a typical question below. Any examples that I can find in textbooks/online consider the problems of a positive definite matrix and I am unable to see how this changes the nature of the problem.

Consider the problem of maximising $\displaystyle x^T Ax $ subject to $\displaystyle x^Tx=1 $ where x is a column vector of n variables and A is a negative-definite, symmetric n-by-n constant matrix. Show that the maximum value of the objective function is equal to the largest eigenvalue of A.

Any help would be much appreciated