# Dimension of space composed of powers of a matrix

1. Nov 29, 2011

### Grothard

Let A be an n*n matrix.
Consider the space $span \{ I, A, A^2, A^3, ... \} .$
How would one show that the dimension of the space never exceeds n?
I feel like the answer lies somewhere near the Cayley-Hamilton theorem, but I can't quite grasp it.

2. Nov 30, 2011

### micromass

Staff Emeritus
The answer is indeed given by Cayley-Hamilton.

Basically, what you must prove is that every matrix like An, An+1, etc. can be written as a linear combination of the matrices I,A,...,An-1.

Now, what does Cayley-Hamilton say?? Doesn't this theorem answer our question?

How would you use Cayley-Hamilton to write An as a linear combination of I,A,...,An-1?