# The real positive definite completion problem: cycle completability

### About this Title

**Wayne W. Barrett**, **Charles R. Johnson** and **Raphael Loewy**

Publication: Memoirs of the American Mathematical Society

Publication Year
1996: Volume 122, Number 584

ISBNs: 978-0-8218-0473-5 (print); 978-1-4704-0169-6 (online)

DOI: http://dx.doi.org/10.1090/memo/0584

MathSciNet review: 1342017

MSC (1991): Primary 15A99; Secondary 05C50, 15A48

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Given a partial symmetric matrix, the positive
definite completion problem asks if the unspecified entries in the
matrix can be chosen so as to make the resulting matrix positive
definite. Applications include probability and statistics, image
enhancement, systems engineering, geophysics, and mathematical
programming. The positive definite completion problem can also be viewed
as a mechanism for addressing a fundamental problem in Euclidean
geometry: which potential geometric configurations of vectors (i.e.,
configurations with angles between some vectors specified) are
realizable in a Euclidean space. The positions of the specified entries
in a partial matrix are naturally described by a graph. The question of
existence of a positive definite completion was previously solved
completely for the restrictive class of chordal graphs and this work
solves the problem for the class of cycle completable graphs, a
significant generalization of chordal graphs. These are the graphs for
which knowledge of completability for induced cycles (and cliques)
implies completability of partial symmetric matrices with the given
graph.

Readership

Graduate students and research mathematicians interested
in graphs and matrices.

### Table of Contents

**Chapters**

- 1. Introduction
- 2. Graph theory concepts
- 3. Basic facts about the positive definite completion problem
- 4. Examples
- 5. Main result
- 6. The implication $(1.0’) \Rightarrow (1.1)$
- 7. The implication $(1.1) \Rightarrow (1.2)$
- 8. The implication $(1.2) \Rightarrow (1.3)$
- 9. The implication $(1.3) \Rightarrow (1.0)$