An invariant of 2 by 2 matrices

Abstract

Let W be the space of 2×2 matrices over a field K. Let f be any linear function on W
that kills scalar matrices. Let A ∈ W anddefine f_k(A) = f(A^k). Then the quantity f_k+1(A)/f(A)
is invariant under conjugation and moreover f_k+1(A)/f(A) = trace S^kA, where S^kA is the k-th
symmetric power of A, that is, the matrix giving the action of A on homogeneous polynomials of
degree k.