I know by the Taylor expansion $f(x,y)$ that in order for the origin to be a minimum point, $f_{xx}$ and $f_{yy}$ have to be both positive. Which I know how to prove. I also know other methods like implicit differentiation and Lagrange multipliers.
Now what I having trouble proving is,
Showing that a unit vector $(x,y)$ that maximizes or minimizes $(x \ y )M(x \ y )^T$ is an eigenvector of $M$, which is a real, symmetric matrix.
How do I prove that?