HELP!!! MATRIX QUESTION!?

Question

I'm trying to figure out how to raise a matrix to a power BY HAND. I guess its more of a syntax question though:

When you see [A]^3, does this mean raise the matrix to the third power? I know it may seem like a stupid question but there are a lot of little syntax rules to remember and I just want to make sure this is what it is saying. Thanks all!

Answer 1

Yes, A^3 means A * A * A. You multiply the matrix by itself, then multiply that result by the original matrix. It can take a while to do this by hand when you have a matrix, but it's doable.



There ARE some shortcuts if your matrix and/or your exponent is really big. These typically involve "diagonalization". In general, a diagonal matrix "D" (where every entry above and below the main diagonal is 0) raised to n is just the matrix where the main diagonal is (d11)^n, (d22)^n, (d33)^n, etc. You can make use of this to diagonalize a given matrix:

http://en.wikipedia.org/wiki/Diagonalizable_matrix



But again, this really only saves you time when when you have a really big exponent to work with. Something like A^3 can usually be done by hand in a shorter amount of time. If you're just beginning to work with matrices, you probably won't end up using diagonalization anytime soon.

Answer 2

Yes, that would mean to multiply matrix AxAxA.

Answer 3

Yes, just do A*A*A... you need to find B=(A*A) first then multiply the new matrix (B) by A.