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Cambridge Tracts in Mathematics #116: Matrices of Sign-Solvable Linear Systemsby Richard A. Brualdi
Synopses & Reviews
In a sign-solvable linear system, the signs of the coefficients determine the signs of some entries in the solution. This type of system is part of a larger study that helps researchers understand if properties of a matrix can be determined from combinatorial arrangements of its elements. In this book, the authors present the diffuse body of literature on sign-solvability as a coherent whole for the first time, giving many new results and proofs and establishing many new connections. Brualdi and Shader describe and comment on algorithms implicit in many of the proofs and their complexity. The book is self-contained, assuming familiarity only with elementary linear algebra and graph theory. Intended primarily for researchers in combinatorics and linear algebra, it should also be of interest to computer scientists, economists, physicists, chemists, and engineers.
Book News Annotation:
A unified and self-contained presentation of sign-solvability, revealing it as a beautiful interplay among combinatorics (especially graph theory), linear algebra, and theoretical computer science (combinatorial algorithms). The organization of the material affords new connections among various results in the literature, as well as giving new results and new and simpler proofs of previously established results. A noteworthy feature is the explicit description of algorithms that are implicit in many of the proofs, with commentary on their complexity. There are chapter bibliographies as well as a master bibliography. Primarily for researchers in combinatorics and linear algebra, but also of interest to theoretical computer scientists, economists, physicists, chemists, and engineers.
Annotation c. Book News, Inc., Portland, OR (booknews.com)
In a sign-solvable linear system, some qualities of the solution are determined solely by the signs of the coefficients. Paul Samuelson extolled the use of such investigations in economics; applications have also been found in chemistry and physics. This book presents the diffuse field of sign-solvability as a coherent whole for the first time, giving many new results and establishing numerous new connections. Algorithms implicit in many of the proofs are explicitly described.Intended primarily for researchers in combinatorics and linear algebra, this self-contained account will also interest computer scientists, economists, physicists, chemists and engineers.
Presents sign-solvability and its applications as a coherent whole; includes many new results.
Table of Contents
1. Sign-solvability; 2. L-matrices; 3. Sign-solvability and digraphs; 4. S-matrices; 5. Beyond S*-matrices; 6. SNS-matrices; 7. S2NS-matrices; 8. External properties of L-Matrices; 9. The inverse sign pattern graph; 10. Sign stability; 11. Related topics.
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