Matrix Powers

There is a relatively simple way of raising a matrix to any power, even negative or decimal-powers:

Diagonalize A so that A = UPU⁻¹, and P is a diagonal matrix. Then Aⁿ = UPⁿU⁻¹ for any n. (I assume you know how to do this since you've been working with matrix exponents.)

For large matrices, you should not do this if you are not calculating a negative or decimal power, because the most efficient way of diagonalizing a matrix as of today is to, well, multiply it. You are basically doing the work twice. Of course, if you need several powers, diagonalizing is efficient.

And yeah, matrix exponential is

,

which is the exact same definition of e^x for any number x too, if you use Taylor series.

So, say we want to find 10^x. We can then express this in this way, as you did with the matrix:



This means that



So, as you see, there is nothing different about matrices, compared to real number when using this equation. Do you consider this weird, or is this just an unexpected result? Both sums converge, as k! grows faster than any polynomial or combination of them.

Noah