22. Diagonalization and Powers of A
OK. Shall we start? This is the second lecture on eigenvalues. So the first lecture was -- reached the key equation, A x equal lambda x. x is the eigenvector and lambda's the eigenvalue. Now to use that. And the, the good way to, after we've found -- so, so job one is to find the eigenvalues and find the eigenvectors. Now after we've found them, what do we do with them? Well, the good way to see that is diagonalize the matrix. So the matrix is A. And I want to show -- first of all, this is like the basic fact. This, this formula. That's, that's the key to today's lecture. This matrix A, I put its eigenvectors in the columns of a matrix S. So S will be the eigenvector matrix. And I want to look at this magic combination S inverse A S. So can I show you how that -- what happens there? And notice, there's an S inverse. We have to be able to invert this eigenvector matrix S. So for that, we need n independent eigenvectors. So that's the, that...