Note Set 20b: Diagonalization is just the best

The monks help us see that if a matrix $\mathbf{A}$ has a linearly independent set of eigenvectors then we can, amazingly, factorize $\mathbf{A}$ through diagonalization: $\mathbf{A} = \mathbf{S}\mathbf{\Lambda}\mathbf{S}^{-1}$. We derive the factorization, talk about how $\mathbf{A}\vec{x}$ really works through basis transformation, and how we can easily find arbitrary powers of $\mathbf{A}. We examine a simple example 2 by 2 to deliver the happiness in full. Basking is allowed and encouraged.
Best consumed by 2016/11/07