The maximal determinant and subdeterminants of \(\pm\)1 matrices.
type: Item
instance of: scholarly article
title: The maximal determinant and subdeterminants of \(\pm\)1 matrices.
zbMATH Open document ID: 1048.15008
DOI: 10.1016/S0024-3795(03)00584-6
author: Tianbing Xia
author: Jennifer Seberry
author: Christos Koukouvinos
author: Marilena Mitrouli
published in: Linear Algebra and its Applications
publication date: 2003-11-19T00:00:00Z
review text: A matrix of order \(n\) is said to be an Hadamard matrix if all its entries are \(\pm 1\) and its distinct row and column vectors are orthogonal. The authors investigate \(\pm 1\)-matrices with maximal absolute values of determinants, especially Hadamard matrices, and their submatrices. It is proved that if \(H\) is a \(12 \times 12\)-Hadamard matrix then a square \(\pm 1\)-matrix of size 5 with a maximal absolute value of the determinant can be embedded to \(H\). Several conjectures concerning the numbers that can appear as maximal values of determinants for \(\pm 1\)-matrices are posed and verified for small values of \(n\). Among them there is a conjecture that the determinant of a \(\pm 1 \)-matrix of order \(n\) is a multiple of \(2^{n-1}\). Principal minors occurring in a completely pivoted \(\pm 1\)-matrix are studied. In particular, it is proved that every Hadamard matrix of order \(n\geq 4\) contains a submatrix which can be obtained from the matrix \(A= E_{11}+E_{12}+E_{13}+E_{14}+E_{21}-E_{22}+E_{23}-E_{24}+ E_{31}+E_{32}-E_{33}-E_{34}+E_{41}-E_{42}-E_{43}+E_{44}\) by permutation of rows/columns and multiplication of rows/columns with \(\pm 1\). It is shown that every completely pivoted Hadamard matrix contains \(A\) in its left upper corner. An algorithm to compute \((n-j)\times (n-j)\) minors of Hadamard matrices of order \(n\) is presented and these minors are determined for \(j=1,2,3,4\).
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 65F40
zbMATH DE Number: 2005987
zbMATH Keywords: determinants
zbMATH Keywords: Hadamard matrices
zbMATH Keywords: \(\pm 1\)-matrices
zbMATH Keywords: algorithm
zbMATH Keywords: minors
zbMATH Keywords: subdeterminants
zbMATH Keywords: completely pivoted matrix
reviewed by: Alexander Guterman
MaRDI profile type: Publication
cites work: The maximum determinant of 21\(\times 21\) \((+1,-1)\)-matrices and D-optimal designs
cites work: Q4503102
cites work: Pivot size in Gaussian elimination
cites work: An Algorithm for the Permanent of Circulant Matrices
cites work: Growth in Gaussian Elimination
cites work: On the complete pivoting conjecture for a hadamard matrix of order 12
cites work: A counterexample to a hadamard matrix pivot conjecture
cites work: Q3665259
cites work: Determinantenabschätzungen für binäre Matrizen. (Estimation of determinants for binary matrices)
cites work: Determinantenabschätzung für binäre Matrizen mit \(n\equiv 3\bmod 4\)
cites work: Q4198777
cites work: Q4379760
cites work: Q4500613
cites work: An algorithm to find formulae and values of minors for Hadamard matrices
cites work: An algorithm to find formulae and values of minors for Hadamard matrices. II
cites work: The exact D-optimal first order saturated design with 17 observations
cites work: Q3858393
cites work: Combinatorics. Room squares, sum-free sets, Hadamard matrices
cites work: On Hadamard's inequality for the determinants of order non-divisible by 4
full work available at URL: s0024-3795(03)00584-6
OpenAlex ID: W1976208484
description: scientific article; zbMATH DE number 2005987