ELA, Volume 13, pp. 146-152, June 2005, abstract.

An Invariant of 2x2 Matrices

Jose Luis Cisneros-Molina

Let W be the space of 2x2 matrices over a field K. 
Let f be any linear function on W that kills scalar 
matrices. Let A belong to W and define 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.