On the dimension of twisted centralizer codes

Given a field F, a scalar λ∈F and a matrix A∈Fn×n, the twisted centralizer code CF(A,λ):={B∈Fn×n|AB−λBA=0} is a linear code of length n2 over F. When A is cyclic and λ≠0 we prove that dim⁡CF(A,λ)=deg⁡(gcd⁡(cA(t),λncA(λ−1t))) where cA(t) denotes the characteristic polynomial of A. We also show how CF(A,λ) decomposes, and we estimate the probability that CF(A,λ) is nonzero when |F| is finite. Finally, we prove dim⁡CF(A,λ)⩽n2/2 for λ∉{0,1} and 'almost all' n×n matrices A over F.