Memoirs of the American Mathematical Society 1996; 69 pp; softcover Volume: 122 ISBN10: 0821804731 ISBN13: 9780821804735 List Price: US$40 Individual Members: US$24 Institutional Members: US$32 Order Code: MEMO/122/584
 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  Introduction
 Graph theory concepts
 Basic facts about the positive definite completion problem
 Examples
 Main result
 The implication \((1.0')\Rightarrow (1.1)\)
 The implication \((1.1)\Rightarrow (1.2)\)
 The implication \((1.2)\Rightarrow (1.3)\)
 The implication \((1.3)\Rightarrow (1.0)\)
