$A^k = I$ implies diagonalizable?

Question

If $A$ is a square complex matrix with $A^k = I$ (where $I$ is the identity matrix of the same size as $A$) for some positive integer $k$, does it follow that $A$ is diagonalizable?

score 1 · Answer 1 · answered Aug 30 '15 at 08:05

1

Yes. Notice that the polynomial with simple roots $x^k-1$ annihilates the matrix $A$ so $A$ is diagonalizable.

answered Aug 30 '15 at 08:05

$A^k = I$ implies diagonalizable?

1 Answers1