Note Set 23b: Positive Definite Matrices in the Wild

We show how certain polynomial functions can be gainfully re-written in the form $\vec{x^{\rm T}} \mathbf{A} \vec{x}$ with $\mathbf{A}$ chosen to be a real, symmetric matrix. For example, $f(x_1,x_2) = \left[ x_1 x_2 \right] \left[\begin{array}{cc} a & b \\ c & d \end{array}\right] \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right]$. Good things follow. For example, if $\mathbf{A}$ is a PDM then $f$ has a simple minimum at $\vec{x}=0$. We also test for maxima, saddles, and horses.
Best consumed by 2016/11/28