Signings

Let A be an m by n matrix. Recall that A is an L-matrix if and only if every matrix in the qualitative class Q(A) has linearly independent rows. If A is an L-matrix, then every matrix obtained from A by appending column vectors is also an L-matrix. If A is an L-matrix and each of the m by n – 1 matrices obtained from A by deleting A column is not an L-matrix, then A is called A barely L-matrix [1]. Thus A barely L-matrix is an L-matrix in which every column is essential. If A is an L-matrix, then we can obtain A barely L-matrix by deleting certain columns of A. An SNS-matrix, that is, A square L-matrix, is A barely L-matrix. But there are barely L-matrices which are not square. The 3 by 4 matrix (1.10) is an L-matrix, and it follows from Theorem 1.2.5 that each of its submatrices of order 3 is not an SNS-matrix. Hence (1.10) is A barely L-matrix.

A signing of order k: is A nonzero (0, 1, – 1)-diagonal matrix of order k. A strict signing is A signing that is invertible. Let D = diag(d1, d2,…, dk) be A signing of order k with diagonal entries d1, d2,…,dk. If k = m, then the matrix DA is A row signing of the matrix A, and if D is A strict signing, then DA is A strict row signing of A.

A problem in economics

Qualitative economics is usually considered to have originated with the work of Samuelson [11, Chapter III] who discussed the possibility of determining unambiguously the qualitative behavior of solution values of A system of equations. In his pioneering paper [6] (see also [4] and [7, 8]) Lancaster put it this way: Economists believed for A very long time, and most economists would still hope it to be so, that A considerable body of sensible economic propositions could be expressed in A qualitative way, that is, in A form in which the algebraic sign of some effect is predicted from A knowledge of the signs, only, of the relevant structural parameters of the system.

Consider the following example, similar to one discussed in Samuelson [11], of A market for A product, say bananas, where the price and quantity are determined by the intersection of its supply and demand curves. We introduce A shift parameter α into the demand curve, and assume that an increase in A shifts the demand curve upward and to the right. For instance, α might represent people's taste for bananas, and as people's taste for bananas increases so does their demand for bananas. Let S(p) denote the number x of bananas that farmers will produce if the price per banana is p. Simple economic principles tell us that as the price p increases farmers will supply more bananas. This gives A supply curve as indicated in Figure 1.1.

Let D(p,a) denote the number x of bananas that consumers will demand if the price per banana is p and people's taste for bananas is α.

The possibility of writing this book occurred to us in the late fall of 1991 when we were both participating in the program on Applied Linear Algebra at the Institute for Mathematics and its Applications (IMA) in Minnesota. A few years earlier we had been attracted to the subject of sign-solvability because of the beautiful interplay it afforded among linear algebra, combinatorics, and theoretical computer science (combinatorial algorithms). The subject, begun in 1947 by the economist P. Samuelson, was developed from various perspectives in the linear algebra, combinatorics, and economics literature. We thought that it would be A worthwhile project to organize the subject and to give A unified and self-contained presentation. Because there were no previous books or even survey papers on the subject, the tasks of deciding what was fundamental and how the material should be ordered for exposition had to be thought out very carefully. Our organization of the material has resulted in new connections among various results in the literature. In addition, many new results and many new and simpler proofs of previously established results are given throughout the book. We began the book in earnest in early 1992 and completed approximately three quarters of it while in residence at the IMA. After we returned to our home institutions, with the other duties that that entails, it was difficult to find the time for completing the book.

One of the features of this book is that we have explicitly described algorithms that are implicit in many of the proofs and have commented on their complexity. Throughout we have given credit for results that have previously occurred in the literature.