Eigenvalue of Matrix Powers

Theorem
Let $A$ be a square matrix.

Let $\lambda$ be an eigenvalue of $A$ and $\mathbf v$ be the corresponding eigenvector.

Then:

holds for each positive integer $n$.

Proof
We will proove the statement by mathematical induction.

Clearly, the statement holds for $n=1$.

Induction hypothesis:

Suppose that $A^n {\mathbf v} = \lambda^n {\mathbf v}$ holds for some positive integer $n$.

Then: