# Homework Help: Eigenvalue proof

1. Oct 24, 2011

### nickw00tz

1. The problem statement, all variables and given/known data
Proof: Prove that if A is an nxn (square mtx) such that A^2=A, then A has 0 or 1 as an eigenvalue.

3. The attempt at a solution
A=A^2
A^2-A=0
A(A-I)=0
A=0 or A=1
and then plugging the A solutions in to the characteristic equation and solving for λ

2. Oct 24, 2011

### lanedance

start with eigenvalue $lambda$ with a correpsonding eigenvector u.

Is u also an eigenvector of A^2?

if so what is the corresponding eigenvalue?

Last edited: Oct 24, 2011
3. Oct 24, 2011

### nickw00tz

can we assume, for the proof, the eigenvalues are both equal to λ?

4. Oct 24, 2011

### lanedance

no, don't assume, but can you show it?

Also though I get your meaning please be explicit in you question (eg. what do you mean by "both")

5. Oct 24, 2011

### nickw00tz

Sorry about that, what I meant was could we associate λ as an eigenvector for A and A^2. For example:

If Au=λu
then (A^2)u=λu, where u=/=0

6. Oct 24, 2011

### lurflurf

^What do you mean both there are n eigenvalues.
Are you working over a splitting field?
Sketch of proof
1)since A=A2
A and A2 have the same eigenvalues
2)find out when A and A2 have the same eigenvalues

7. Oct 24, 2011

### HallsofIvy

If v is an eigenvector of A with eigenvalue $\lambda$, then $A^2v= A(Av)= A(\lambda v)= \lambda Av= \text{what?}$