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.

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  • anon  On December 12, 2011 at 10:09 am

    I think that this is not true for the field with 2 elements, since 1=-1

    • nbloomf  On December 12, 2011 at 11:13 am

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

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