L-matrix

Matrices playing a central role in the study of qualitative economics and first defined by P.A. Samuelson [a6]. A real -matrix is an -matrix provided every matrix with the same sign pattern as has linearly independent rows. For example,

are -matrices. A linear system of equations, , is called sign-solvable provided the signs of the entries in any solution can be determined knowing only the signs of the entries in and . If the linear system is sign-solvable, then is an -matrix. General references for this area include [a1], [a3] and [a4].

The study of -matrices has included characterizations of structural properties, classification of subclasses as well as interrelationships with other discrete structures. For example, two subclasses of -matrices which arise are that of the barely -matrices and the totally -matrices.

An -matrix is a barely -matrix provided that is an -matrix but if any column of it is deleted, the resulting matrix is not an -matrix.

An -matrix is a totally -matrix provided that each -submatrix of is an -matrix.

The two matrices and above are examples of barely -matrices. The matrix is also a totally -matrix but is not since its -submatrix made up of the first three columns is not an -matrix. The matrix

is a totally -matrix.

The property of being a barely -matrix, or a totally -matrix, imposes restrictions on the number of columns. If is an barely -matrix, then the number of columns is at most ; further, if has only non-negative entries, then the number of columns is at most

If is an totally -matrix, then the number of columns is at most . It has been shown that the set of all by totally -matrices can be obtained from the matrix above by performing certain extension operations on successively [a2].

An important subclass of the -matrices for which there exist a great deal of literature is that of the square -matrices, which are also called sign-non-singular matrices.

References

[a1]	L. Bassett, J. Maybee, J. Quirk, "Qualitative economics and the scope of the correspondence principle" Econometrica , 36 (1968) pp. 544–563
[a2]	R.A. Brualdi, K.L. Chavey, B.L. Shader, "Rectangular L-matrices" Linear Algebra Appl. , 196 (1994) pp. 37–61
[a3]	R.A. Brualdi, B.L. Shader, "Matrices of sign solvable systems" , Tracts in Math. , 116 , Cambridge Univ. Press (1995)
[a4]	V. Klee, R. Ladner, R. Manber, "Sign-solvability revisited" Linear Algebra Appl. , 59 (1984) pp. 131–157
[a5]	R. Manber, "Graph-theoretical approach to qualitative solvability of linear systems" Linear Algebra Appl. , 48 (1982) pp. 131–157
[a6]	P.A. Samuelson, "Foundations of economic analysis" , Economic Studies , 80 , Harvard Univ. Press (1947)