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Totally Nonnegative Matrices$
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Shaun M. Fallat and Charles R. Johnson

Print publication date: 2011

Print ISBN-13: 9780691121574

Published to University Press Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691121574.001.0001

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date: 13 December 2017

Extensions and Completions

Extensions and Completions

Chapter:
(p.185) Chapter Nine Extensions and Completions
Source:
Totally Nonnegative Matrices
Author(s):

Shaun M. Fallat

Charles R. Johnson

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691121574.003.0010

This chapter explores the relatively modern idea of matrix completion problems for TN matrices, and reviews a variety of construction, extension, and completion ideas for both TP and TN matrices. The property that a matrix be TP is sufficiently strong that, at first glance, construction seems even more difficult than recognition. Of course, the elementary bidiagonal factorization provides an easy way simply to write down an example, but with this factorization it is very difficult to “design” many entries of the resulting matrix. This chapter presents a strategy to remedy this, before turning to completions and partial TN matrices, MLBC graphs (special chordal graphs), and TN completions. It also includes a section on single entry perturbations, known as retractions, which turn out to be a useful tool for other problems on TN matrices.

Keywords:   extensions, completions, matrix completion problems, constructions, TN completions, line insertion, partial TN matrices, MLBC graphs, TN perturbations, retractions

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