Homework Help: Linear Algebra -- Projection matrix question

1. Apr 25, 2016

Physgeek64

1. The problem statement, all variables and given/known data
Let A be an n×n matrix which has the property that A^2 =A.
(i) Write down the most general polynomial in A

2. Relevant equations

3. The attempt at a solution
My biggest problem is that I don't even understand what the question is asking

Is it just sum (alphaA^n)=0

but A^n=A

sum(alpha A)=0 ?

I know its not the equation to find the eigenvalues as that follows, and I'm fine with that

Av=pv where p are the eigenvalues, and v the corresponding eigenvectors
A^2v=Apv
Av=p(Av)
pv=p^2v
v(p)(p-1)=0 and hence p= 0 or 1

But I just don't understand the first bit

2. Apr 25, 2016

Simon Bridge

The absolute most general polynomial, degree N, in some arbitrary A would be $P_N(A)=\sum_{n=0}^N a_nA^n$ wouldn't it?

3. Apr 25, 2016

Physgeek64

Well since A is a projection matrix, surely that would imply than $A^n=A$ and hence $P_N(A)=\sum_{n=0}^N a_nA^n$ -> $P_N(A)=\sum_{n=0}^N a_nA$

4. Apr 25, 2016

Simon Bridge

i.e. the most general polynomial is 1st order. Well done.

5. Apr 25, 2016

LCKurtz

Wouldn't you want to include $A^0 = I$?

6. Apr 26, 2016

Physgeek64

Yes, yes I would ;) Thank you