Chapter 7 - Eigenvalues and Eigenvectors - 7.3 Diagonalization - Problems - Page 460: 25

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We have: $$A=SDS^{-1}$$ then $$A^2=(SD^2S^{-1})=(SDS^{-1})(SDS^{-1})$$ For $m \gt 2$, we obtain $A^m=SD^mS^{-1} \\ A^{m+1}=A.A^m=(SDS^{-1})(SD^mS^{-1})=SD^{m+1}S^{-1}$ Hence, for every $k=1,2,3.... \rightarrow A^k=SD^kS^{-1}=$