# Eigenvectors and using them in matrix algebra.

1. Mar 27, 2005

### Zhiv

Hi.

Marix A=

|1 1 0 |
|0 2 0 |
|2 1-1 |

Has three eigenvectors [1,1,1]^T, [1,0,1]^T and [0,0,1]^T, By using this knowledge solve A^11.

Ok, solving A^11 is rather easy with any decent calculator, or even with pen , paper and some time, but how on earth I'm supposed to benefit from thoose eigenvectors?

Thank you.

2. Mar 27, 2005

### Muzza

Look up "diagonalization".

Let P be the 3x3 matrix whose columns consists of the eigenvectors of A. Then P^-1AP is a diagonal matrix (where the entries are the eigenvalues of the eigenvectors). If D is that diagonal matrix, we have P^-1AP = D <=> A = PDP^-1, so that A^m = P * D^m * P^-1 for natural m (the last step can be proven with induction). But calculating D^m is easy, just raise each non-zero entry to the power of m. Then it's just a matter of working out what P^-1 is, and then multiplying the matrices.

Last edited: Mar 27, 2005