Interior eigenvectors of symmetric matrices are saddle points

Eigenpairs of symmetric matrices are intimately related to optimization and critical points, with the eigenvectors being critical points of the Rayleigh quotient. In optimization settings, the type of critical point (minimum, maximum, saddle point) is an important feature in designing and understanding algorithms.

This write up characterizes the nature of all critical points (eigenvectors), with the main result being that all interior eigenvectors are saddle points.