Does there exist a definition for matrix exponentiation?
If we have, say, an integer, one can define $A^B$ as follows:
$$\prod_{n = 1}^B A$$
We can define exponentials of fractions as a power of a radical, and we even have the following definition of the exponential:
$$e^z = \sum_{n = 0}^\infty \frac{z^n}{n!}$$
which comes from a Taylor Series for the function $\exp(z)$. Now, a problem seems to arise when we attempt to calculate $\exp(A)$, where $A$ is an $n$ x $n$ (square) matrix. We cannot define it as multiplication a "matrix number of times" as this makes no sense. The only reasonable definition that could work is the latter definition (the infinite series):
$$e^A = 1 + A + \frac{AA}{2!} + \frac{AAA}{3!} + \frac{AAAA}{4!} + \cdots$$
where we can define matrix exponentiation to the power of an integer, which is all that is required here. We know that $e^x$ will converge absolutely for all complex numbers, but do we know that this is true for matrices?
Can this "matrix sum" diverge, and are there ways to test divergence/convergence when a matrix is applied?
Or is this concept of "matrix divergence" not well defined?
Best Answer
Yes, the exponential of a matrix can be defined by that Taylor series, and it is a very useful thing. See e.g. Wikipedia The proof that it converges is not difficult, using any sub-multiplicative matrix norm. If $\| \cdot \|$ is such a norm, then any power series $\sum_n c_n A^n$ converges whenever the real series $\sum_n |c_n| \|A\|^n$ converges.