Note Set 22a: Symmetry, the Spectral Theorem, and More Happiness

Diagonalization goes to 11 when we take on symmetric matrices. The eigenvectors are now always guaranteed to be linearly independent and, almost unbelievably, they form an orthogonal basis for $R^n$. Real symmetric matrices also have real eigenvalues (no rotations). Even more amazingly, diagonalization now works so that we see $\mathbf{A}$ as a sum of weighted, outer-product-based projection operators. We present the basic story and examine our simple example.
Best consumed by 2016/11/09