Eigenvalue of the Power of a Matrix

Question

Let $A$ be an n×n matrix with eigenvalues $\lambda_1,\dots,\lambda_n$. Show that $\lambda_1^k,\dots,\lambda_n^k$ are the eigenvalues of $A^k$.

I don't know where to start.

Answer 1

Let $v_1,v_2,\cdots, v_n$ be eigenvectors of matrix $A$, where $v_i$ corresponding to eigenvalue $\lambda_i$. Then by definition of eigenvector and eigenvalue:

$$Av_i=\lambda_iv$$

So:

$$A^nv_i=A^{n-1}(Av_i)=A^{n-1}\lambda_iv_i=\lambda_iA^{n-1}v_i=\\=\lambda_iA^{n-2}(Av_i)=\lambda_iA^{n-2}\lambda_iv_i=\lambda_i^{2}A^{n-2}v_i=\cdots=\lambda_{i}^{n}v_i$$