Matrix 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.)

There are several problems to this. I came across matrix exponentials as a way to solve first order linear systems (of ODEs). By diagonalizing the matrix, I'd have to compute its eigenvalues and eigenvectors, which then makes the process kind of useless because using the eigenvalues and eigvenvectors, I could compute the answer directly. Also, I'm not sure when this happens, but I do know that some matrices aren't diagonalizable, and I'm not sure whether or not all compositions (for lack of a better term) of n x 1 vectors into n x n matrices have inverses.

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.

It was an unexpected result, since I previously didn't think about diagonalization. I'm still not sure how this is fitting and if it varies continuously, since I lack results. One difficulty with that I face with this problem is that my graphing calculator can't handle matrix exponentials, non-integer powers, and negative powers, so I can't compute any results.

If we consider the identity matrix, the identity matrix to any power must equal itself. After all, that's the definition of the identity matrix. However, if we tried to "diagonalize it", we'd get:

Aⁿ = UPⁿU⁻¹

[1, 0; 0, 1]^n = [1, 0; 0, 1] [1, 0; 0, 1]^n [1, 0; 0, 1]

Or equivalently,

[1, 0; 0, 1]^n = [0, 1; 1, 0] [1, 0; 0, 1]^n [0, 1; 1, 0]

I'm not sure how this even changes anything.

What happens when the eigenvalues are complex? Is there a geometric interpretation of this? In general, is there anything useful that this is used for?