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Let $A \in \mathbb{C}(n,n)$ and $k \geq 2$ be an integer such that $$A \sim A^k$$. Show that if $A$ is non-singular then each eigenvalue of $A$ is a root of unity.

Attempt: Since $A \sim A^k$, $$PA = A^kP$$ where $P$ is an invertible matrix. Since $A$ is invertible, $0$ cannot be an eigenvalue of $A$. Suppose $$Av = \lambda v \quad v \neq 0$$ then $$PAv = \lambda Pv$$ $$\therefore A^k(Pv) = \lambda (Pv) $$

which implies that $Pv$ is an eigenvector of $A^k$. But the eigenvalues of $A^k$ are $\lambda^k$ $$\therefore \lambda^k=\lambda$$ which gives the conclusion required.

My questions is: Is the logic correct? If so, Am I missing any details? If not, then how could I approach this?

Thanks!

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  • $\begingroup$ The phrase "but the eigenvalues of $A^k$ are $\lambda^k$" seems imprecise. $\endgroup$ Aug 10, 2013 at 5:25
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    $\begingroup$ Your last conclusion is invalid. $A$ may have several eigenvalues, $\lambda, \mu$, and you might have that $\mu^k=\lambda$. $\endgroup$
    – vadim123
    Aug 10, 2013 at 5:25

1 Answer 1

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You are correct in noting that similar matrices have the same set of eigenvalues. However, given this, you cannot conclude that $\lambda = \lambda^k$. However, you can conclude that $\lambda^k$ is an eigenvalue if $\lambda$ is.

So suppose there are $n$ eigenvalues total (there must be finitely many). Take the set $\{ \lambda , \lambda^k , \lambda^{k^2} , \lambda^{k^3}, \dots, \lambda^{k^n} \}$. This has $n+1$ elements, and each element is an eigenvalue, so by the pidgeonhole principle, we must repeat an element twice.

So $\lambda^{k^m} = \lambda^{k^i}$ for some $1 \le i < m \le n$.

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  • $\begingroup$ How did you get the set of $\lbrace \lambda,...,\lambda^{k^{n}}\rbrace$ $\endgroup$
    – AAP
    Aug 10, 2013 at 7:02
  • $\begingroup$ @AAP Since you know that $\lambda$ is an eigenvalue, you know that $\lambda^k$ is as well, and so is $(\lambda^k)^k = \lambda^{k^2}$ and so on... $\endgroup$
    – A.S
    Aug 10, 2013 at 17:17
  • $\begingroup$ @A.S The lower bound of $i$ should be $0$. $\endgroup$
    – user371231
    Jan 10 at 5:05

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