Pseudo-centrosymmetric matrices, with applications to counting perfect matchings

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• 作者：Christopher R.H. Hanusa
• 关键词：Centrosymmetric ; Anti-involutory ; Pseudo-centrosymmetric ; Determinant ; Alternating centrosymmetric matrix ; Jockusch ; 2-even symmetric graph ; Kasteleyn-Percus ; Domino tiling ; Aztec diamond ; Aztec pillow
• 刊名：Linear Algebra and its Applications
• 出版年：2007
• 出版时间：1 December 2007
• 年：2007
• 卷：427
• 期：2-3
• 页码：206-217
• 全文大小：208 K

文摘

We consider square matrices A that commute with a fixed square matrix K, both with entries in a field F not of characteristic 2. When K² = I, Tao and Yasuda defined A to be generalized centrosymmetric with respect to K. When K² = −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.