zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Matrices of sign-solvable linear systems. (English) Zbl 0833.15002
Cambridge Tracts in Mathematics. 116. Cambridge: Cambridge Univ. Press. ix, 298 p. £30.00; $ 49.95 (1995).

The subject of sign-solvability of a linear system was begun in 1947 by the economist P. Samuelson and later developed from various perspectives in the linear algebra, combinatorics and economics literature. The present book organizes the subject and gives it a unified and self- contained presentation.

The sign-solvability of a linear system implies that the signs of the entries of the solution are determined solely on the basis of the signs of the coefficients of the system. Sign-solvability is part of a larger study which seeks to understand the special circumstances under which an algebraic, analytic or geometric property of a matrix can be determined from the combinatorial arrangement of the positive, negative, and zero elements of the matrix. Thus these are properties shared by all members of a qualitative class of matrices.

Several classes of matrices arise in this way, notably sign-nonsingular matrices, L-matrices, S-matrices, and sign-stable matrices. (The essential idea of a sign-nonsingular matrix arose in a different context in a 1963 paper by P. W. Kasteleyn.) The large and diffuse body of literature connected with sign-solvability is presented as a coherent whole for the first time in this book displaying it as a beautiful interplay between linear algebra, combinatorics (especially graph theory) and theoretical computer science (combinatorial algorithms).

The book should be of interest not only to researchers in combinatorics and linear algebra, but also to economists, theoretical computer scientists, physicists, chemists and engineers.

15A06Linear equations (linear algebra)
05C50Graphs and linear algebra
15-02Research monographs (linear algebra)
68R05Combinatorics in connection with computer science