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Linear Algebra

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Moderate

Eigenvalues of Matrix Powers

LINALG-HGFZXE

Let $A$ be an $n \times n$ matrix and let $k$ be a positive integer.

What is the relationship between the eigenvalues of $A$ and the eigenvalues of $A^k$?

A

If $\lambda$ is an eigenvalue of $A$, then $\lambda$ is an eigenvalue of $A^k$.

B

If $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$.

C

If $\lambda$ is an eigenvalue of $A$, then $k \lambda$ is an eigenvalue of $A^k$.

D

If $\lambda$ is an eigenvalue of $A$, then $\lambda^{1/k}$ is an eigenvalue of $A^k$.

E

None of the above.