The Real Positive Definite Completion Problem: Cycle Completability (Memoirs of the AMS) (Memoirs of the American Mathematical Society) - Softcover

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

Softcover

ISBN 10: 0821804731 ISBN 13: 9780821804735

Publisher: American Mathematical Society, 1996

Synopsis

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
graphs
for which knowledge of completability for induced cycles (and cliques) implies completability of partial symmetric matrices with the given graph.

Edition details

• Publisher: American Mathematical Society
• Publication date: 1996
• Language: English
• ISBN 10: 0821804731
• ISBN 13: 9780821804735
• Binding: Paperback
• Number of pages: 69