On the dimension of twisted centralizer codes

Given a field F , a scalar λ ∈ F and a matrix A ∈ F, the twisted centralizer code CF (A, λ) := {B ∈ F | AB − λBA = 0} is a linear code of length n. When A is cyclic and λ 6= 0 we prove that dimCF (A, λ) = deg(gcd(cA(t), λcA(λt))) 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 dimCF (A, λ) 6 n /2 for λ 6∈ {0, 1} and ‘almost all’ matrices A.