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Can you find a minimal polynomial of $A^n$ if you know the minimal polynomial of $A$?

  • I'm talking about minimal polynomials of matrices.
  • I'm asking in the general sort of way, I know that in some cases you can use algebraic tricks
    or some other cleverness.
  • $n$ is just some integer, it's not connected to the matrix in any way.
  • 0
    @OlivierBégassat $n$ is just some integer, I edited my question to make it clear. – 2012-11-12

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This can be answered in general by examining the Jordan structure of $A$. Recall that the eigenvalues of $A^n$ are precisely $\lambda^n$ for each eigenvalue $\lambda$ of $A$. Multiplicity is preserved.

If the minimal polynomial contains a factor of the form $x^k$ then this is transformed to $x^{\left\lceil\frac{k}{n}\right\rceil}$.

Factors of the form $(x-\lambda)^k$ are preserved as $(x-\lambda^n)^k$. If it happens that we have $\lambda_1^n = \lambda_2^n$ for distinct eigenvalues $\lambda_1$ and $\lambda_2$, then we take on the factor with the largest exponent.