Higher Power of a square Matrix

[Guidenable]

Homework Statement

Given the matrix A=
-1/5 7/5
-3/5 -4/5

find A⁴³.

The Attempt at a Solution

It's obvious that I can't go and actually compute A⁴³ so there must be a more elegant way of doing this. The only notes I have on the subject is A^k=P^-1D^kP, where D is a diagonal matrix. However, I have no clue what P is supposed to to, nor why this would work in the first place.

 

[phyzguy]

You're on the right track. If you've studied eigenvalues and eigenvectors, you should be able to calculate D and P.

 

[Guidenable]

I've never heard of eigenvalues or eigenvectors. I'm in a college level Linear Algebra I class, so I don't know if I should or not.

 

[Mark44]

You need to have some understanding of eigenvalues and eigenvectors to be able to diagonalize a matrix. In your formula, the columns of matrix P are the eigenvectors of matrix A, and P^-1 is the inverse of P. Matrix D is a diagonal matrix whose entries are the eigenvalues of A.

If you're expected to work a problem like this, there must be similar problems in your textbook, and some presentation of these ideas must have been given in class.

 

[phyzguy]

I think you should have. Wikipedia has a good entry on how to do it here:

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

 

[Guidenable]

Ah ok, it's quite possible that the prof mentioned it but I missed it. Thanks for the link, I'll put it to good use.

 

[HallsofIvy]

Also, the eigenvalues for this particular matrix are complex numbers. That's going to make calculating the 43 power even more complicated. Fortunately, they both have modulus 1.