Inequalities in products of minors of totally nonnegative matrices. (English) Zbl 1066.05089
A matrix is called totally nonnegative if each of its minors is nonnegative and totally positive if the minors are positive. A more traditional terminology called these matrices, respectively, totally positive and strictly totally positive, but those denominations are becoming more used. Totally nonnegative matrices are important in many branches of mathematics, as CAGD, probabilty, economics, combinatorics, graph theory, etc.
The author of the present paper considers these matrices from the point of view of counting (weighted) paths in directed graphs and uses these techniques to study inequalities in products of minors. More precisely, he gives a characterization of inequalities of the form \[ \Delta_{I,I'}\Delta_{K,K'}\leq \Delta_{J,J'}\Delta_{L,L'} \] which hold for all totally nonnegative matrices. Here \(\Delta_{I,I'}\) denotes the determinant of the submatrix formed with rows indexed \(I\) and columns indexed \(I'\) of a matrix \(A\) and similarly \(\Delta_{K,K'}, \Delta_{J,J'},\Delta_{L,L'}\).
The author of the present paper considers these matrices from the point of view of counting (weighted) paths in directed graphs and uses these techniques to study inequalities in products of minors. More precisely, he gives a characterization of inequalities of the form \[ \Delta_{I,I'}\Delta_{K,K'}\leq \Delta_{J,J'}\Delta_{L,L'} \] which hold for all totally nonnegative matrices. Here \(\Delta_{I,I'}\) denotes the determinant of the submatrix formed with rows indexed \(I\) and columns indexed \(I'\) of a matrix \(A\) and similarly \(\Delta_{K,K'}, \Delta_{J,J'},\Delta_{L,L'}\).
Reviewer: Mariano Gasca (Zaragoza)
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A45 | Miscellaneous inequalities involving matrices |
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