Determinants of some special matrices over commutative finite chain rings

Abstract

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

Determinants of Arrowhead Matrices over Finite Commutative Chain Rings

“…Diagonal matrices are interesting subfamilies of the ones in [3]. The enumeration of diagonal matrices over FCCRs of a fixed determinant are presented in [8] and applied in the study of the determinant of some circulant matrices over FCCRs.…”

“…Arrowhead matrices have applications in various fields, e.g., wireless communications in [15], eigenvalue decompositions of some matrices in [16], the study of directed multigraphs and hub-directed multigraphs in [12], and the study of disordered quantum spins in [4]. As a generalization of [8], the enumeration of arrowhead matrices with prescribed determinant over a FCCR is investigated in the following set up. For a FCCR R, let U (R) denote the set of units in R and let Z(R) denote the set of zero-divisors in R. Let A n (R) denote the set of n × n arrowhead matrices over R. It is not difficult to see that A n (R) is a group under addition and…”