# General truth about elements of a matrix to the n-th power?

1. Jan 19, 2012

### nonequilibrium

Say we have a matrix P with eigenvalues $\lambda_1, \cdots, \lambda_n$ (possibly some are the same) and P can be diagonalized, then we can always say that the element on the a'th row and b'th column of P^n is equal to $P^n(a,b) = \sum_{i = 1}^n \alpha_i \lambda_i^n$ with $\alpha_i$ independent of n (but dependent on a and b).

Correct? And I don't think the conditions can be weakened?

2. Jan 19, 2012

### micromass

Staff Emeritus
Seems indeed correct.

3. Jan 19, 2012

### nonequilibrium

I find it to be rather pretty :)

4. Jan 19, 2012

### micromass

Staff Emeritus
Indeed, it IS pretty!!