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 ifB = 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.

I had. Damn!

I have corrected it. It was in among the SVD posts, specifically “the SVD generalizes eigenstructure”. if someone else had written my post, I think I would have caught that. Unfortunately, when I read my own stuff, I sometimes read what I meant to write, instead of what I did write. (That’s how I learned to let another grad student take a test before I handed it out to my students. He’d have to read what I wrote; take it myself, and I don’t even stop to read the questions!)

As I said, if A is symmetric, we may choose P orthogonal. More generally, if A is hermitian, we may choose P unitary. In ultimate generality, we may choose P unitary if and only if A is normal.

I have even shown you that we cannot always have P unitary. The counterexample in the schur’s lemma post was specifically a non-normal matrix which could be diagonalized, but whose eigenvectors were not orthogonal, i.e. whose eigenvector matrix was not, and could not be made, unitary.

FWIW, for any 3D rotation matrix – which is orthogonal, hence normal – we may choose P unitary but not orthogonal: even though the rotation matrix is real, its eigenvector matrix is complex and can only be made unitary.

schur’s lemma: any matrix is unitarily similar to an upper triangular

i bumped into someone last night who asked me about schur’s lemma, something about bringing a matrix to triangular form. i’ve spent so much time looking at diagonalizng things that i didn’t appreciate schur’s lemma, and it deserves to be appreciated.
it says that we can bring any (complex) matrix A to upper triangular form using a unitary similarity transform. in this form, the restriction to “unitary” is a bonus: a perfectly useful but weaker statement is that any matrix is similar to an upper triangular matrix.
now, we’re usually interested in diagonalizing a matrix. when can we go that far?
easy: that upper triangular matrix is in fact diagonal iff the original matrix A is normal; that is, iff A commutes with its conjugate transpose:
A \ A^{\dagger } = A^{\dagger }\ A.

so, any normal matrix can be diagonalized; furthermore, the similarity transform is unitary.

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