The real positive definite completion problem : cycle completability / Wayne W. Barrett, Charles R. Johnson, Raphael Loewy.

Format:

Book

Author/Creator:

Barrett, Wayne W. (Wayne Walton), 1948- author.

Johnson, Charles R., author.

Loewy, Raphael, 1943- author.

Series:

Memoirs of the American Mathematical Society ; Volume 122, Number 584.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 122, Number 584

Language:

English

Subjects (All):

Graph theory.

Matrices.

Physical Description:

1 online resource (82 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island, United States : American Mathematical Society, 1996.

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""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') â?? (1.1)""; ""7 The Implication (1.1) â?? (1.2)""; ""8 The Implication (1.2) â?? (1.3)""; ""9 The Implication (1.3) â?? (1.0)""; ""References""

Notes:

"July 1996, volume 122, number 584 (fourth of 5 numbers)"--Cover.

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0169-X