# Properties of $n \times n$ complex matrix with $A^m = I$

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?

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