Let . We define

It is known that this series converges. Prove that

**Solution**

We triangulise the matrix , that is where is an invertible matrix and is an upper triangular. This is possible since our matrix is over and thus its characteristic polynomial splits. Let be its eigenvalues. Then we note that is upper triangular with in its diagonal. Hence is also upper triangular with in its diagonal. Hence

However and forall . Thus and finally

The interested reader may also take a look at the exercise here at mathimatikoi.org .

Since the space of all complex matrices is a finite-dimensional vector space, all norms define the same topology. So we can take a sub-multiplicative norm, that is, a norm such that . (For example, we can take to be the operator norm on E.) As a finite dimensional vector space, E is complete, so it’s enough to show normal convergence. We have that, for each integer ,

and we know that, for each real number , the series converges (it defines the exponential function). Therefore, for any , the series converges.

(We also got the additional result that . )