Matrix Powers

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.

Well, if you cannot diagonalizable the matrix, you will not have enough eigenvectors and -values. Therefore you cannot solve through eigenvalues and -vectors, and you need to either represent the problem in another way (probably without matrices), or use some other property to bypass the lack of eigenvectors and -values.

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.

Use Mathematica, Maple or Frink? Or Python, if you want to. (I usually use Python, actually).

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.

Is there any surprising effect here which surprises you? Of course the identity matrix of any power is the identity matrix.

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?

Oh, yeah, forgot to tell about that.
Nothing special, actually. Just work with it as you usually work with imaginary numbers.

I haven't really seen many visual examples of complex matrices. However, the eigenvalues of rotation-matrices are imaginary. In the 2D-plane, you can imagine that it works as magnetism, and that the eigenvectors and eigenvalues are the magnetic field which rotates the plane.

Noah