View source for L-matrix

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.

Out of 58 formulas, 58 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
Matrices playing a central role in the study of qualitative economics and first defined by P.A. Samuelson [[#References|[a6]]]. A real $( m \times n )$-matrix $A$ is an $L$-matrix provided every matrix with the same sign pattern as $A$ has linearly independent rows. For example,

\begin{equation*} M = \left( \begin{array} { c c c } { 1 } &amp; { - 1 } &amp; { 0 } \\ { 1 } &amp; { 1 } &amp; { - 1 } \\ { 1 } &amp; { 1 } &amp; { 1 } \end{array} \right) , \quad N = \left( \begin{array} { c c c c } { 1 } &amp; { 1 } &amp; { 1 } &amp; { - 1 } \\ { 1 } &amp; { 1 } &amp; { - 1 } &amp; { 1 } \\ { 1 } &amp; { - 1 } &amp; { 1 } &amp; { 1 } \end{array} \right) \end{equation*}

are $L$-matrices. A linear system of equations, $A x = b$, is called sign-solvable provided the signs of the entries in any solution can be determined knowing only the signs of the entries in $A$ and $b$. If the linear system $A x = b$ is sign-solvable, then $A ^ { T }$ is an $L$-matrix. General references for this area include [[#References|[a1]]], [[#References|[a3]]] and [[#References|[a4]]].

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

An $( m \times n )$-matrix $A$ is a barely $L$-matrix provided that $A$ is an $L$-matrix but if any column of it is deleted, the resulting matrix is not an $L$-matrix.

An $( m \times n )$-matrix $A$ is a totally $L$-matrix provided that each $( m \times m )$-submatrix of $A$ is an $L$-matrix.

The two matrices $M$ and $N$ above are examples of barely $L$-matrices. The matrix $M$ is also a totally $L$-matrix but $N$ is not since its $( 3 \times 3 )$-submatrix made up of the first three columns is not an $L$-matrix. The matrix

\begin{equation*} T = \left( \begin{array} { c c c c } { 1 } &amp; { 1 } &amp; { 1 } &amp; { 0 } \\ { 1 } &amp; { - 1 } &amp; { 0 } &amp; { 1 } \end{array} \right) \end{equation*}

is a $( 2 \times 4 )$ totally $L$-matrix.

The property of being a barely $L$-matrix, or a totally $L$-matrix, imposes restrictions on the number of columns. If $A$ is an $( m \times n )$ barely $L$-matrix, then the number of columns is at most $2 ^ { m - 1 }$; further, if $A$ has only non-negative entries, then the number of columns is at most

\begin{equation*} \left( \begin{array} { c } { m } \\ { \lceil \frac { m + 1 } { 2 } \rceil } \end{array} \right). \end{equation*}

If $A$ is an $( m \times n )$ totally $L$-matrix, then the number of columns is at most $m + 2$. It has been shown that the set of all $m$ by $m + 2$ totally $L$-matrices can be obtained from the matrix $T$ above by performing certain extension operations on $T$ successively [[#References|[a2]]].

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

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