Let be an matrix over such that . Show that can be diagonalized. Is this result true if we replace by an arbitrary field ?

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

## Comments

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

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