How do you calculate a characteristic polynomial of a matrix in GF(2)? I understand the concept of characteristic polynomials in matrices using "regular" math with real numbers, but I'm a bit confused about how it works in GF(2).
Is there a relation between the characteristic polynomial and eigenvectors?
(I understood how to do this about ten years ago but it's been too long and I can't find my work from back then.)
edit: I guess the problem I'm having is that regular math matrices with integer values can have a characteristic polynomial with roots that are neither integer or real numbers, and those numbers make sense; I can't get my head around what happens in GF(2) algebra where similar cases occur. (e.g. characteristic polynomial = $x^3 + x + 1$)
The characteristic polynomial of a matrix $M$ is always $\det (M - tI)$. Over a finite field, the coefficients of this polynomial lie in the same finite field but their roots lie in algebraic extensions of this finite field in general. A finite field $\mathbb{F}_p$ has a unique such extension of order $n$ for each positive integer $n$, denoted $\mathbb{F}_{p^n}$ or, in your notation, $\text{GF}(p^n)$. Any reference on finite fields will explain this.