Pseudo-centrosymmetric matrices, with applications to counting perfect matchings

Simple item page
dc.creatorHanusa, Christopher R. H.
dc.date2006-09-21
dc.date2007-07-09
dc.date.accessioned2026-07-07T08:14:26Z
dc.date.available2026-07-07T08:14:26Z
dc.descriptionWe consider square matrices A that commute with a fixed square matrix K, both with entries in a field F not of characteristic 2. When K^2=I, Tao and Yasuda defined A to be generalized centrosymmetric with respect to K. When K^2=-I, we define A to be pseudo-centrosymmetric with respect to K; we show that the determinant of every even-order pseudo-centrosymmetric matrix is the sum of two squares over F, as long as -1 is not a square in F. When a pseudo-centrosymmetric matrix A contains only integral entries and is pseudo-centrosymmetric with respect to a matrix with rational entries, the determinant of A is the sum of two integral squares. This result, when specialized to when K is the even-order alternating exchange matrix, applies to enumerative combinatorics. Using solely matrix-based methods, we reprove a weak form of Jockusch's theorem for enumerating perfect matchings of 2-even symmetric graphs. As a corollary, we reprove that the number of domino tilings of regions known as Aztec diamonds and Aztec pillows is a sum of two integral squares.
dc.descriptionv1: Preprint; 11 pages, 7 figures. v2: Preprint; 15 pages, 7 figures. Reworked so that linear algebraic results are over a field not of characteristic 2, not over the real numbers. Accepted, Linear Algebra and its Applications
dc.identifierhttps://arxiv.org/abs/math/0609622
dc.identifierhttp://arxiv.org/abs/math/0609622
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/133119
dc.subjectCombinatorics
dc.subjectRings and Algebras
dc.subjectPrimary: 05C75, 15A15, 15A21; Secondary: 05B20, 05B45, 05C50, 15A23
dc.titlePseudo-centrosymmetric matrices, with applications to counting perfect matchings
dc.typetext

Files

Collections

arXiv