On the cardinality of matrices with prescribed rank and partial trace over a finite field

Abstract

Let F be the finite field of order q and M(n, n, r, F ) be the set of n x n matrices of rank r over the field F . For alpha E F and A E M(n, n, F ), let Z alpha A,r = {X X E M(n, n, r, F ) Tr(AX) AX ) = alpha } . In this article, we solve the problem of determining the cardinality of Z alpha A,r . We also solve the generalization of the problem to rectangular matrices. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.