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Principally Unimodular Skew-Symmetric Matrices

André Bouchet, W. H. Cunningham, J. F. Geelen
1998 Combinatorica  
A square matrix is principally unimodular if every principal submatrix has determinant 0 or 1. Let A be a symmetric (0; 1)-matrix, with a zero diagonal.  ...  A PU-orientation of A is a skew-symmetric signing of A that is PU. If A 0 is a PUorientation of A, then, by a certain decomposition of A, we can construct every PU-orientation of A from A 0 .  ...  Introduction A square matrix A is called principally unimodular (PU) if every nonsingular principal submatrix is unimodular (that is, has determinant 1).  ... 
doi:10.1007/s004930050033 fatcat:gfaf7reyinhavcj3ui747gughm

Integral Solutions of Linear Complementarity Problems

William H. Cunningham, James F. Geelen
1998 Mathematics of Operations Research  
We summarize what is known about principally unimodular symmetric and skew-symmetric matrices.  ...  These matrices, called principally unimodular matrices, are those for which every principal nonsingular submatrix is unimodular.  ...  We shall denote by M X] the principal submatrix M X; X]. Obviously symmetric and skew-symmetric matrices are rank-symmetric.  ... 
doi:10.1287/moor.23.1.61 fatcat:vps63izvezdm7amd7m7g5zqyb4

Skew-symmetric matrix polynomials and their Smith forms

D. Steven Mackey, Niloufer Mackey, Christian Mehl, Volker Mehrmann
2013 Linear Algebra and its Applications  
Restricting the class of equivalence transformations to unimodular congruences, a Smith-like skew-symmetric canonical form for skew-symmetric matrix polynomials is also obtained.  ...  These results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew-symmetric matrices, as well as the existence of structured linearizations  ...  We thank Leiba Rodman for bringing to our attention the application to symmetric factorizations of skew-symmetric rational matrices, and we thank André Ran and Leiba Rodman for helpful discussions.  ... 
doi:10.1016/j.laa.2013.02.010 fatcat:f5b7oa66u5cjfc65sycxr5ease

Page 567 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 39, Issue 9 [page]

1933 American Mathematical Society. Bulletin of the American Mathematical Society  
For if 2A =S+Q, where S is symmetric and Q is skew, then 2AT=S—Q, so that 2[A + AT — S] = 0, 2[4 — AT-Q] = 0. 3. Unimodular Matrices.  ...  A matrix S such that S’=S is called symmetric. A matrix Q such that Q' = —Q is called skew.  ... 

Skew circulant quadratic forms

Dennis Garbanati, Robert C. Thompson
1972 Journal of Number Theory  
This paper investigates positive definite unimodular quadratic forms in n variables with rational integer coefficients and a skew circulant as the coefficient matrix.  ...  It is shown for n < 13 that every such form is in the principal class, but that this no longer holds for n = 14. It is also shown that such forms can never be even.  ...  Let A and B be it x n positive definite symmetric unimodular matrices with rational integers as entries.  ... 
doi:10.1016/0022-314x(72)90028-5 fatcat:mj6ft7hccncujnkuhrmecka4me

On unimodular tournaments

Wiam Belkouche, Abderrahim Boussaïri, Abdelhak Chaïchaâ, Soufiane Lakhlifi
2021 Linear Algebra and its Applications  
A tournament is unimodular if the determinant of its skew-adjacency matrix is 1. In this paper, we give some properties and constructions of unimodular tournaments.  ...  A unimodular tournament T with skew-adjacency matrix S is invertible if S^-1 is the skew-adjacency matrix of a tournament. A spectral characterization of invertible tournaments is given.  ...  The inverse S −1 of S is a unimodular skew-symmetric integral matrix, but its off-diagonal entries are not necessarily from {−1, 1}.  ... 
doi:10.1016/j.laa.2021.09.014 fatcat:2t6lnpber5hr3dtkxlpupf6yc4

A note on the spectra of certain skew-symmetric {1,0,−1}-matrices

Gerhard Wesp
2002 Discrete Mathematics  
We characterize skew-symmetric {1; 0; −1}-matrices with a certain combinatorial property. In particular, we exhibit several equivalent descriptions of this property.  ...  These results allow characterizations of unimodular orientations of the complete graph, of rank 2 chirotopes, and of a class of multipartite oriented graphs.  ...  A is called a principal unimodular (PU) matrix if the principal minors det A| F are in {1; 0; −1} for all F ⊆ E, where A| F =(a xy ) x;y∈F : PU matrices have received some attention in the literature,  ... 
doi:10.1016/s0012-365x(02)00402-8 fatcat:dz64y5zfzremdjtq3sckedhxxm

A normal form for Riemann matrices

A. A. Albert
1965 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Two Riemann matrices co and coo are said to be equivalent if (2) holds with A a unimodular integral matrix. The relation of equivalence is also easily seen to be an equivalence relation.  ...  A matrix co having p rows and 2p columns of complex number elements is called a Riemann matrix of genus p if there exists a rational 2^-rowed skew matrix C such that is positive definite Hermitian.  ...  Indeed the matrix (5) C, J» E ) \-E 0/ is the usual canonical form for the skew matrix C under unimodular congruence, and Co is a principal matrix of co 0 .  ... 
doi:10.4153/cjm-1965-097-0 fatcat:rt327yiwczhs7o5r7s4qjaoqtq

Matrices with elements in a principal ideal ring

C. C. MacDuffee
1933 Bulletin of the American Mathematical Society  
By this I mean that the basic theorems of number theory, such as unique factorization into primes, hold for a principal ideal ring, while the concept of principal ideal ring is sufficiently general to  ...  The mathematical system which seems most satisfactory as an abstraction of the system of rational integers is the principal ideal ring.  ...  For if 2A = S + Q, where 5 is symmetric and Q is skew, then 2AT = S -Q, so that 2[A +A^ -S] = 0, 2[A -At -Q] = 0. 3. Unimodular Matrices.  ... 
doi:10.1090/s0002-9904-1933-05681-1 fatcat:vdm6573nubdp3la744xkp2hiou

Skew-symmetric matrices and their principal minors [article]

A. Boussairi, B. Chergui
2015 arXiv   pre-print
Loewy [5] for skew-symmetric matrices whose all off-diagonal entries are nonzero.  ...  Given a positive integer k, what is the relationship between two matrices A=(a_ij)_i,j∈ V, B=(b_ij)_i,j∈ V with entries in K and such that (A[ X])=(B[ X]) for any subset X of V of size at most k ?  ...  It follows that if A, B are two skew-symmetric dense matrices have equal corresponding principal minors of order at most 4, then they are both principally unimodular or not.  ... 
arXiv:1403.0095v2 fatcat:uim4fyi4nvhbbbhbuqhr2w4acm

Page 857 of Mathematical Reviews Vol. 58, Issue 2 [page]

1979 Mathematical Reviews  
Further, for a general principal ideal ring, the problem is shown to depend on whether matrices in SL(3, R) can be expressed as products of symmetric matrices.  ...  The author studies the ques- tion of when a matrix in SL(m, R) may be expressed as a product of symmetric matrices in SL({n, R), where R is a principal ideal ring.  ... 

Determinants of matrices related to the Pascal triangle

Roland Bacher
2002 Journal de Théorie des Nombres de Bordeaux  
Skew-symmetric matrices Given an arbitrary sequence a = (ao, al, ... ) with ao = 0, the matrices are skew-symmetric.  ...  This submatrix is given by The set of sequences associated to unimodular skew-symmetric matrices thus consists of integral sequences and has the structure of a tree.  ... 
doi:10.5802/jtnb.344 fatcat:w6oq3273abgynpyi5avk5mz7fy

Page 4637 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
and skew-symmetric determinant.  ...  submatrices of normal, Hermitian and symmetric matrices.  ... 

Four-dimensional lattice rules generated by skew-circulant matrices

J. N. Lyness, T. Sørevik
2003 Mathematics of Computation  
We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree d.  ...  We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules.  ...  In terms of generator matrices, symmetric copies of a lattice may be created by postmultiplying by permutation matrices and sign change matrices.  ... 
doi:10.1090/s0025-5718-03-01534-5 fatcat:bxlttbnvabhftcyjkkfnd5oumq

A Generalization of Tutte's Characterization of Totally Unimodular Matrices

J.F. Geelen
1997 Journal of combinatorial theory. Series B (Print)  
Principal unimodularity was originally studied with regard to skew-symmetric matrices; see [2, 4, 5] ; here we consider symmetric matrices.  ...  Theorem 2.2 (Bouchet [3] Bouchet proved that Theorem 2.2 also holds for skew-symmetric matrices.  ... 
doi:10.1006/jctb.1997.1751 fatcat:edrh7npr3vblvekopvg4r6lbby
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