Suppose a $n$ by $n$ matrix $A$ is non-singular. Suppose $v$ is a eigenvector of $A^m$, where $m$ is a positive integer. Is it ture that $v$ is also an eigenvector of $A$? If it is true, how to prove it?
2026-04-01 14:25:02.1775053502
Is an eigenvector of the power of a non-singular matrix also an eigenvector of that non-sigular matrix?
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It turns out that it is false. Take $A=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$. What are the eigenvectors of $A^2$? Are they eigenvectors of $A$ too?