Science by Katharine

A sampling from my Mathematical Methods for Physics course, showing the stationary points of a quadratic form $f\left({\bf x}\right) = {\bf x}^T A {\bf x}$ subject to the normalization constraint $\Vert {\bf x} \Vert = 1$ . The constrained stationary points ${\bf v}$ are normalized eigenvectors of the symmetric eigenvalue problem $A{\bf v}=\lambda {\bf v}$ , and the values of $f$ at the stationary points are the eigenvalues for those eigenvectors. This connection between symmetric eigensystems and constrained optimization gives a nice geometrical interpretation of the eigenvectors, and connects to how Schrödinger guessed his quantum-mechanical wave equation as well as Rayleigh quotient iteration for approximation of eigenvalues and Rayleigh-Schrödinger perturbation theory.