----- > [!proposition] Proposition. ([[eigenvalues of matrix power]]) > If $A \in \mathbb{F^{}}^{N \times N}$ has [[eigenvalue]]s $\lambda_{1},\dots,\lambda_{N}$ (possibly with repeats), then $A^{k}$ has [[eigenvalue]]s $\lambda_{1}^{k},\dots,\lambda_{N}^{k}$. > > [!proof]- Proof. ([[eigenvalues of matrix power]]) >Let $v$ be an arbitrary [[eigenvector]] of $A$ with [[eigenvalue]] $\lambda$. We have $\begin{align} A^{k}\lambda = A^{k-1}Av = \lambda A^{k-1}v = \dots = \lambda^{k}v. \end{align}$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```