Note Set 23a: Totally, for sure, Positive Definite Matrices

We venture into a world of elegant square matrices, the ones of Positive Definiteness. For our purposes, Positive Definite Matrices (PDMs) are real, symmetric, square matrices that have only positive eigenvalues ($\lambda_i > 0\ \forall\ i$). We develop a test for PDMness and find a surprising connection between pivots and eigenvalues: their signs must match up. Theory first in this episode, and then example usages in the ones following.
Best consumed by 2016/11/28