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

Kumar Balasubramanian; Krishna Kaipa; Himanshi Khurana

paper

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

arXiv:2407.14273

Abstract

Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n \times n$ matrices of rank $r$ over the field $F$ . For $α \in F$ and $A\in \M(n,F)$, let $$Z^α_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=α\right \}.$$ In this article, we solve the problem of determining the cardinality of $Z_{A, r}^{α}$ . We also solve the generalization of the problem to rectangular matrices.