## Corrections

So far, there are only two that I know of. One is here in “attitude & transition matrices”; the other is here in “the SVD generalizes eigenstructure”.

I was reading yet another book on PCA / FA last night, and I came across a definition saying that two matrices A and B were similar if$B = P^T\ A\ P$. Of course, I objected that P needed to be orthogonal, and we couldn’t guarantee that in general. This morning, out with one of the cats – and, therefore, thinking instead of computing or writing! – I observed that the author had made that mistake because the matrices $X^T\ X$ and $X\ X^T$ are symmetric, and for symmetric matrices it is certainly true that P may be chosen orthogonal. He had been careless because within the realm of PCA / FA, we are only finding eigendecompositions of symmetric matrices. Within the realm of PCA / FA, we may take P to be orthogonal, because we’re looking at $X^T\ X$ and/or $X\ X^T$ when we’re not using the SVD.

It was a while after that I began to wonder if I had made the very same mistake.