Power of matrix and power of eigenvalue

[bmanbs2]

Assuming that k\geq0,

How does one prove that when A has an eigenvaule \lambda that A^{k} has an eigenvalue \lambda^{k}?

 

[disregardthat]

This is pretty straight-forward. Av=\lambda v for some vector v, so try to calculate A^kv.

 

[bmanbs2]

Matrix raised to certain power has exponent raised to same power proof.

Homework Statement

Prove that when A has an eigenvaule \lambda that A^{k} has an eigenvalue \lambda^{k}?

Homework Equations

None

The Attempt at a Solution

Tried to show that A^{k}X^{k} = \lambda^{k}X^{k}, but X^{k} isn't possible as X is a vector.

 

[Dick]

Realizing A^k*X^k makes no sense is a good start. But A^2(X)=A(A(X))=A(lambda*X)=lambda*(A(X))=lambda*(lambda*X)=lambda^2*X makes sense, doesn't it?

 

[bmanbs2]

Yes...yes it does indeed. Thank you