**Memoirs of the American Mathematical Society**

1996;
69 pp;
Softcover

MSC: Primary 05; 15;

Print ISBN: 978-0-8218-0473-5

Product Code: MEMO/122/584

List Price: $40.00

AMS Member Price: $24.00

MAA Member Price: $36.00

**Electronic ISBN: 978-1-4704-0169-6
Product Code: MEMO/122/584.E**

List Price: $40.00

AMS Member Price: $24.00

MAA Member Price: $36.00

# The Real Positive Definite Completion Problem: Cycle Completability

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*Wayne W. Barrett; Charles R. Johnson; Raphael Loewy*

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

# Table of Contents

## The Real Positive Definite Completion Problem: Cycle Completability

- Contents vii8 free
- 1 Introduction 110 free
- 2 Graph Theory Concepts 312 free
- 3 Basic Facts about the Positive Definite Completion Problem 716
- 4 Examples 1221
- 5 Main Result 1726
- 6 The Implication (1.0') ⇒ (1.1) 1928
- 7 The Implication (1.1) ⇒ (1.2) 2433
- 8 The Implication (1.2) ⇒ (1.3) 3039
- 9 The Implication (1.3) ⇒ (1.0) 5867
- References 6877