If $A^m = I_n$ then what can we say about the eigenvalues and diagonalizablity of $A$?

The equation given above is an annihilating polynomial of $A$ and therefore minimal polynomial divides it. Since the roots of the polynomial are distinct in complex field. Hence it is diagonalizable with each eigenvalue being some root of unity. Am I correct?

  • 4
    $\begingroup$ Yes, it is diagonalisable, for the reason you give. $\endgroup$ – Lord Shark the Unknown Mar 16 at 7:13
  • $\begingroup$ Thanks and what about the nature of eigenvalues? @LordSharktheUnknown $\endgroup$ – Devendra Singh Rana Mar 16 at 7:14
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    $\begingroup$ Same : eigenvalues are roots of the minimal polynomial, therefore they must be roots of unity. $\endgroup$ – Ayoub Mar 16 at 7:18

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