In the last post, I asked if there exist square matrices such that . This came up in class just after we discussed the spectral mapping theorem, so I was attempting to use that to attack this, but it turns out (as Shikha pointed out) the easiest way is to just look at the trace of both sides. I still haven’t figured out a proof using spectrums, but Dr. Singh assures us the proof will be straightforward once we see a little more machinery.

I found out something interesting this morning, while browsing through ‘Vector Calculus, Linear Algebra, and Differential Forms’ by Hubbard and Hubbard. Recall that the Frechet derivative of a function at , where are normed linear spaces, is the unique linear transformation satisfying

You can show that if you consider (the set of square matrices of order ) as a normed space and look at the squaring function defined by , then

This shines a new light on the fact that there are no such that , but what are the consequences of this fact on and ?

On a not-so-related note, the same book shows a nice generalization of the derivation of the convergence of geometric series to matrices. If and , then . Pretty neat!

(In all the arguments above, they used the Frobenius norm-- I don't think that matters, however, since is finite-dimensional, so all norms are equivalent.)

Possibly relevant posts:

• Dr. Singh’s Take on the Fourier Unicity Theorem (via Invariant Subspaces) (6/17/2005)
• Inner products (6/7/2006)
• The Gelfand representation theorem (7/4/2008)