Scaffoldings of Totally Positive Matrices and Line Insertion
Abstract
Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work we revisit this problem. First we show that every totally positive matrix can be associated to a certain vertexweighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2109.06369
 arXiv:
 arXiv:2109.06369
 Bibcode:
 2021arXiv210906369C
 Keywords:

 Mathematics  Rings and Algebras;
 15B48;
 15A83