Power of Matrix

How do we compute the power of a matrix? We can use Matrix Multiplication a few times. But do we want to do this 5 times?

Recall the diagonalization . Then . Recall from lemma addition and multiplication preserve diagonalization, we know it is very easy to multiply two diagonal matrices together.

Definition

Let . Then, for any ,

where .

Example

Find a formula for for any positive integer where .

First, diagonalize the matrix if possible. We find that and .
Thus