## discussion

There was a lot going on in the example of the SN decomposition (2 of 3). First off, we found eigenvalues of a non-diagonable matrix A, and constructed a diagonal matrix D from them. Then we found 2 eigenvectors and 1 generalized eigenvector of A, and used them to construct a transition matrix P. We used that transition matrix to go from our diagonal D back to the original basis, and find S similar to D.

So S is diagonable while A is not. And A and S have the same eigenvalues; and **the columns of P should be eigenvectors of S. They are.** The generalized eigenvector that we found for A is an (ordinary) eigenvector of S, but we had to get a generalized eigenvector of A in order to construct S from D.

I wonder. Can I understand the distinction between eigenvectors and generalized eigenvectors by studying S and A? We’ll see.

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