ABSTRACT
The origins of sign pattern matrices are in the book [Sam47] by the Nobel Economics Prize winner P. Samuelson, who pointed to the need to solve certain problems in economics and other areas based only on the signs of the entries of the matrices. The study of sign pattern matrices has become somewhat synonymous with qualitative matrix analysis. The dissertation of C. Eschenbach [Esc87], directed by C.R. Johnson, studied sign pattern matrices that “require” or “allow” certain properties and summarized the work on sign patterns up to that point. In 1995, Richard Brualdi and Bryan Shader produced a thorough treatment [BS95] on sign pattern matrices from the sign-solvability vantage point. There is such a wealth of information contained in [BS95] that it is not possible to represent all of it here. Since 1995 there has been a considerable number of papers on sign patterns and some generalized notions such as ray patterns. We remark that in this chapter we mostly use {+, −, 0} notation for sign patterns, whereas in the literature {1, −1, 0} notation is also commonly used, such as in [BS95]. We further note that because of the interplay between sign pattern matrices and graph theory, the study of sign patterns is regarded as a part of combinatorial matrix theory.
