Is an Eigenvector of A also an Eigenvector of A^2?

[evilpostingmong]

Homework Statement

Prove that if s is an eigenvector of matrix A, then it is also an eigenvector
of matrix A^2.

Homework Equations

As=Ics

The Attempt at a Solution

We know that As=Ics.
So AAs=AIcs
A(As)=A(Ics)
A(As)-A(Ics)=0
A(As-Ics)=0
Since As-Ics=0
A(As-Ics)=A(0)=0

 

[xepma]

Looks about right, but perhaps a bit cumbersome. Just apply the matrix twice to the vector s. In your notation:
(AA)s = A(As) = A(Ics) = Ic(As) = (Ic)^2s
Which is all you need.

 

[CompuChip]

I think it is correct, although admittely I didn't check all of it.
There is an easier way.
From the "relevant equations", you see that you have to show that
A^2 s = I k s
for some number k.

If you start from
A^2 s = A I c s,
can you use two properties of matrices to get the A in front of the s and use A s = I c s again?

[edit]Actually xepma has already given you the complete answer [/edit]

 

[evilpostingmong]

Thanks for the help!