The Commutator of Two Companion Matrices Has Rank at Most Two, with Spectral Consequences over Arbitrary Fields

Abstract

We prove that the commutator [U,Q] = UQ − QU of two n×n companion matrices has rank at most two, over an arbitrary field, by exhibiting an explicit decomposition of [U,Q] as a sum of two rank-one matrices. The mechanism is not special to companion matrices: for any A and any e, the commutator of the two rank-one perturbations A − ue^T and A − ve^T has rank at most two, and the companion case is the specialisation A = N, e = e_n. This contrasts with the classical result of Guralnick and Lanski, who show that with one argument fixed and the other free every rank up to n is attained; the collapse is a property of the constrained pair, not of either matrix.We determine the image exactly. Writing [U,Q] = z e_n^T − w e_{n−1}^T, the linear condition z_n = w_{n−1} is necessary, but sufficient only on the open set w_n ≠ 0; on w_n = 0 the attainable z collapse to a line. This yields an exact count of companion commutators over F_q, namely (q−1)q^(2n−2) + (q^(n−1) − 1)q + 1, strictly smaller than the q^(2n−1) matrices meeting the necessary condition.Combined with the tracelessness of any commutator, the rank bound pins the characteristic polynomial to λ^(n−2)(λ² + c) for a single scalar c, for which we give a closed form in six coefficients of U and Q. The spectrum is {0^(n−2), μ, −μ} with μ² = −c: invariant under λ ↦ −λ, with every power-trace a function of c alone (tr([U,Q]^(2k+1)) = 0 and tr([U,Q]^(2k)) = 2(−c)^k, so 2 tr([U,Q]⁴) = (tr [U,Q]²)²), and [U,Q] similar to −[U,Q] over the base field. Rank two is the exact threshold: at rank three the third elementary symmetric function reappears and the pairing breaks. The argument is characteristic-independent and holds verbatim over GF(2). This upgrades earlier finite-size symbolic verifications (n ≤ 6) of the odd power-trace identities to a theorem valid for all n and all fields.We close with an information-theoretic no-go: because these power-traces are constant on the companion class, they carry no instance-discriminating information and cannot serve as search-pruning signals for problems encoded within that class. The image theorem makes the loss quantitative — the instance space has dimension 2n−1 while every trace factors through one scalar.All claims are re-derived from scratch by an accompanying exact-arithmetic verification script (seed 11, 132 checks), including a constructed witness for the nilpotent branch c = 0 that random sampling does not reach, and exhaustive enumeration over F_q confirming the count.