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