Completely Positive Matrices

By: Abraham Berman, Naomi Shaked-monderer

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Hardcover | 15 April 2003

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216 Pages

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A real matrix is positive semidefinite if it can be decomposed as A=BB'. In some applications the matrix B has to be elementwise nonnegative. If such a matrix exists, A is called completely positive. The smallest number of columns of a nonnegative matrix B such that A=BB' is known as the cp rank of A. This work focuses on necessary conditions and sufficient conditions for complete positivity, as well as bounds for the cp rank. The methods are combinatorial, geometric and algebraic. The required background on nonnegative matrices, cones, graphs and Schur complements is outlined.

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.,." of interest mainly to an applied mathematician but the techniques and the difficulties appeal the to pure mathematician as well.?

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Preface	p. vii
Preliminaries	p. 1
Matrix theoretic background	p. 1
Positive semidefinite matrices	p. 9
Nonnegative matrices and M-matrices	p. 18
Schur complements	p. 24
Graphs	p. 30
Convex cones	p. 41
The PSD completion problem	p. 56
Complete positivity	p. 61
Definition and basic properties	p. 61
Cones of completely positive matrices	p. 71
Small matrices	p. 73
Complete positivity and the comparison matrix	p. 78
Completely positive graphs	p. 83
Completely positive matrices whose graphs are not completely positive	p. 97
Square factorizations	p. 122
Functions of completely positive matrices	p. 129
The CP completion problem	p. 133
CP rank	p. 139
Definition and basic results	p. 139
Completely positive matrices of a given rank	p. 146
Completely positive matrices of a given order	p. 156
When is the cp-rank equal to the rank?	p. 177
Bibliography	p. 193
Notation	p. 199
Index	p. 203
Table of Contents provided by Rittenhouse. All Rights Reserved.

Completely Positive Matrices

At a Glance

Hardcover

Industry Reviews

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How to return your order

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