ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th ILAS Conference

Contributed talks on H- and M-Matrices

Thu 14:30–15:00, Room AV 91.21, Chair: Dawid Janse van Rensburg
On the inverse M-matrix problem for singular, symmetric and irreducible M-matrices
Andrés M. Encinas (UPC Barcelona)
Joint work with Ángeles Carmona (UPC Barcelona); Margarida Mitjana (UPC Barcelona)

In this communication we are concerned with the problem of characterizing when the group inverse of a singular, symmetric and irreducible $M$–matrix is also an $M$–matrix. This question has generated a great amount of works and has only been solved in a few specific cases, mainly involving the combinatorial laplacian of a network. Specifically, Y. Chen et al. established in [2] that the group inverse of a d.d. Jacobi matrix; that is the Laplacian associated with a weighted path, is also an $M$–matrix when its order is lower that $4$ and moreover under severe constraints on the weights. More generally, S.J. Kirkland et al. proved in [3] that the only weighted trees whose Laplacian matrix satisfy the required property are, in addition to the above paths, stars with some constraints on their conductances. Under this constraints they proved that there exist a infinity family of stars with the referred property. The use of Potential Theory techniques, has allowed to consider the problem with generality, avoiding the diagonal dominance hypothesis. Since each singular and symmetric and irreducible $M$–matrix can be identified with a singular and positive semidefinite Schrödinger operator on the connected graph associated with the matrix, the proposed problem consists on determine when the matrix associated with the corresponding Green operator is also an $M$–matrix. With this point of view, we obtained a necessary and sufficient condition for the general case, that allowed us to show that there exist infinitely many singular, symmetric and irreducible Jacobi matrices of any order such that its group inverse is also an $M$–matrix, see [1].

The aim of this work is to analyze the question for singular semidefinite Schrödinger operators on stars, or more generally on trees, and cycles.

This research was supported by the Spanish Research Council under project MTM2014-60450-R.

1. Bendito, E., Carmona, A., Encinas, A.M. y Mitjana, M. The M-matrix inverse problem for singular and symmetric Jacobi matrices, Linear Algebra Appl., 436 (2012), 1090–1098.
2. Chen, Y., Kirkland, S.J. y Neumann, M., Group generalized inverses of M-matrices associated with periodic and nonperiodic Jacobi matrices, Linear and Multilinear Algebra 39 (1995) 325–340.
3. Kirkland, S.J. y Neumann, M., The $M$–matrix group inverse problem for weighted trees, SIAM J. Matrix Anal. Appl. 19 (1998), 226–234.