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:

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