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Principally Unimodular Skew-Symmetric Matrices

1998
*
Combinatorica
*

A square matrix is

doi:10.1007/s004930050033
fatcat:gfaf7reyinhavcj3ui747gughm
*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). ...##
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Integral Solutions of Linear Complementarity Problems

1998
*
Mathematics of Operations Research
*

We summarize what is known about

doi:10.1287/moor.23.1.61
fatcat:vps63izvezdm7amd7m7g5zqyb4
*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*. ...##
###
Skew-symmetric matrix polynomials and their Smith forms

2013
*
Linear Algebra and its Applications
*

Restricting the class of equivalence transformations to

doi:10.1016/j.laa.2013.02.010
fatcat:f5b7oa66u5cjfc65sycxr5ease
*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. ...##
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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*. ...##
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Skew circulant quadratic forms

1972
*
Journal of Number Theory
*

This paper investigates positive definite

doi:10.1016/0022-314x(72)90028-5
fatcat:mj6ft7hccncujnkuhrmecka4me
*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. ...##
###
On unimodular tournaments

2021
*
Linear Algebra and its Applications
*

A tournament is

doi:10.1016/j.laa.2021.09.014
fatcat:2t6lnpber5hr3dtkxlpupf6yc4
*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}. ...##
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A note on the spectra of certain skew-symmetric {1,0,−1}-matrices

2002
*
Discrete Mathematics
*

We characterize

doi:10.1016/s0012-365x(02)00402-8
fatcat:dz64y5zfzremdjtq3sckedhxxm
*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, ...##
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A normal form for Riemann matrices

1965
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

Two Riemann

doi:10.4153/cjm-1965-097-0
fatcat:rt327yiwczhs7o5r7s4qjaoqtq
*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 . ...##
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Matrices with elements in a principal ideal ring

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

doi:10.1090/s0002-9904-1933-05681-1
fatcat:vdm6573nubdp3la744xkp2hiou
*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*. ...##
###
Skew-symmetric matrices and their principal minors
[article]

2015
*
arXiv
*
pre-print

Loewy [5] for

arXiv:1403.0095v2
fatcat:uim4fyi4nvhbbbhbuqhr2w4acm
*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. ...##
###
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. ...##
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Determinants of matrices related to the Pascal triangle

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. ...

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

1996
*
Mathematical Reviews
*

and

*skew*-*symmetric*determinant. ... submatrices of normal, Hermitian and*symmetric**matrices*. ...##
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Four-dimensional lattice rules generated by skew-circulant matrices

2003
*
Mathematics of Computation
*

We present briefly some of the underlying theory of these

doi:10.1090/s0025-5718-03-01534-5
fatcat:bxlttbnvabhftcyjkkfnd5oumq
*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*. ...##
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A Generalization of Tutte's Characterization of Totally Unimodular Matrices

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*. ...

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