Matrices of Sign-Solvable Linear Systems (Cambridge Tracts in Mathematics, Series Number 116)

I'm not really a mathematician at heart. I use math and enjoy using it, but never really learned to love it for its own sake. In other words, this book wasn't written for me. I work with combinatoric problems a fair bit, and hoped that sign-solvable systems (SSS) would be directly applicable to some of my work. I also look at questions where sloppy data prevent exact solutions - if a technique promises at least partial answers when precise answers aren't available, it has my interest. The SSS premise is that, in some cases, I can tell whether parts of my answer are positive, negative, or zero, knowing only whether the inputs were positive, negative, or zero. The simplified character of the questions and answers could possibly work well in special computing environments that I use. I really wanted to like this book and its content. I just wasn't able to connect these abstractions to my real world, though. The focus of this book is on the abstract structures behind SSS, and on formal proofs about specific features of special cases. This book does have a practical side. It specifies a number of algorithms for handling sign-solvable systems of various sorts. It would take a lot of effort to reduce these algorithms to practice, but the information is all there. As books of pure advanced math go, this one seems relatively clear and approachable. Perhaps a more astute reader than me can connect SSS to problems of practical interest. Perhaps, some day, I'll take the time to work through this book in detail. These days, though, I can't spend a lot of time away from the problems I need to solve. Building up a working knowledge of SSS from this book would take intense effort. I really have to put my effort elsewhere. 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. That It Might Be Worthwhile And Possible To Investigate Such Linear Systems Was Recognised By Samuelson In His Classic Book Foundations Of Economic Analysis. 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. 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 Combinatorics And Linear Algebra. One Of The Features Of This Book Is That Algorithms That Are Implicit In Many Of The Proofs Have Been Explicitly Described And Their Complexity Has Been Commented On. Richard A. Brualdi, Bryan L. Shader. Includes Bibliographical References (p. 289-293) And Index.

Presents sign-solvability and its applications as a coherent whole; includes many new results.

Booknews

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, 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.