Note Set 24a: Singular Value Decomposition and Scottish fashion

We arrive at Singular Value Decomposition. The monks see our happiness over the factorization of real, symmetric matrices, $\mathbf{A} = \mathbf{Q} \mathbf{\Lambda} \mathbf{Q}^{\rm T}$, and begin to chant "S-V-D" over and over. It's fun to start with, then a little annoying, so we get on with it. The big deal: All real matrices can be expressed as $\mathbf{A} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^{\rm T}$ where $\mathbf{U}$ and $\mathbf{V}$ are basis transformations and $\mathbf{\Sigma}$ is an $\mathbf{A}$-shaped matrix containing stretching/shrinking factors on its main diagonal. You decide: One of the Great Matrix Factorizations, or The Greatest Matrix Factorization?
Best consumed by 2016/11/28