Glasnik Matematicki, Vol. 53, No. 1 (2018), 51-71.

A COMBINATORIAL INTERPRETATION OF THE LDU-DECOMPOSITION OF TOTALLY POSITIVE MATRICES AND THEIR INVERSES

Muhammad ElGebali and Nermine El-Sissi

Mathematics and Actuarial Science Department, The American University in Cairo, 11 853 Cairo, Egypt
e-mail: m.elgebali@aucegypt.edu
e-mail: nelsissi@aucegypt.edu

Abstract.   We study the combinatorial description of the LDU-decomposition of totally positive matrices. We give a description of the lower triangular L, the diagonal D, and the upper triangular U matrices of the LDU-decomposition of totally positive matrices in terms of the combinatorial structure of essential planar networks described by Fomin and Zelevinsky [5]. Similarly, we find a combinatorial description of the inverses of these matrices. In addition, we provide recursive formulae for computing the L, D, and U matrices of a totally positive matrix.

2010 Mathematics Subject Classification.   15A23, 05C50.

Key words and phrases.   Totally positive matrices, LDU factorization, planar networks.


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DOI: 10.3336/gm.53.1.05


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