## Any matrix A such that A³ = A can be diagonalized

Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ such that $A^3 = A$. Show that $A$ can be diagonalized. Is this result true if we replace $\mathbb{C}$ by an arbitrary field $F$?

Note that the minimal polynomial of $A$ divides $x^3-x = x(x+1)(x-1)$. By Corollary 25 in D&F, $A$ is similar to a diagonal matrix $D$, and moreover the diagonal entries of $D$ are either 0, 1, or -1. Since this factorization of $x^3-x$ holds over any field $F$, in fact $A$ is diagonalizable over any field.

That isn’t a problem. The minimal polynomial still has distinct roots all equal to 0 or $1 = -1$.