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) |
L-matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-matrix&oldid=49877