> It sounds like you're assuming diagonalizability, which is stronger than "eigen values defined over the field". > Even over the complex numbers there are mat

> It sounds like you're assuming diagonalizability, which is stronger than "eigen values defined over the field". > Even over the complex numbers there are matrices that aren't diagonalizable, like [[2,1],[0,2]]. While that's completely true, and I'm glad someone said it, it's not much of an issue for 2×2 matrices. You can tell quickly when it's happened, because the characteristic polynomial is a square (itself easily tested by computing the gcd with the derivative … or just because it's easy to recognize when a quadratic is a perfect square) but the matrix isn't scalar; and then we have that the semisimple part S is scalar, the nilpotent part N satisfies N^2 = 0, and so (S + N)^n equals S^n + n S N^{n - 1} for n > 1.