A note on completing quasi-distance and distance matrices

Yulin Zhang,J M S Simões-Pereira,A Veloso Da Costa

doi:10.1515/spma-2019-0016

Yulin Zhang, J M S Simões-Pereira + Show 1 more

Open Access

https://doi.org/10.1515/spma-2019-0016

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Abstract

Abstract We give a necessary and sufficient condition for the existence of a quasi-distance matrix where some positive off-diagonal entries have been prescribed. Moreover, we give an algorithm for obtaining such a matrix. We analyze also the case where distances instead of quasi-distances are considered.

Highlights

A distance matrix is a square matrix D whose entries are non-negative real numbers which satisfy the following three conditions, the third one being called the triangle inequality:∀i : di,i = ; ∀i, k : di,k = dk,i; ∀i, j, k : di,k ≤ di,j + dj,k
We assume that the entries di,k are positive for i ≠ k, as is usually done; nally, we point out that, if we drop the symmetry in this de nition, D will be called a quasi-distance matrix
We prove a necessary and su cient condition for the following problem to have a solution: Consider an n × n matrix W of real numbers with a main diagonal of zeros and with a few more positive entries prescribed

Summary

Introduction

A distance matrix is a square matrix D whose entries are non-negative real numbers which satisfy the following three conditions, the third one being called the triangle inequality:. We assume that the entries di,k are positive for i ≠ k, as is usually done; nally, we point out that, if we drop the symmetry in this de nition, D will be called a quasi-distance matrix. An interesting application is their use in a biologic model to reconstruct phylogenetic trees from matrices whose entries represent certain genetic distances among biological species (see [2]). We prove a necessary and su cient condition for the following problem to have a solution: Consider an n × n matrix W of real numbers with a main diagonal of zeros and with a few more positive entries prescribed. Choose the remaining entries so that the matrix M we obtain is a quasi-distance matrix

Main result

Start with the following matrix