Completely Positive Matrices, Graphs with no long odd cycles and graphs with no short odd cycles

Avi Berman

Technion-Israel Institute of Technology

February 6,
4:15pm
refreshments at 3:45pm
2-338

ABSTRACT 

A matrix is completely positive if it can be factored as A=BB^T where B is elementwise nonnegative. Such matrices arise in block designs and in statistics and are related to copositive matrices. Clearly if A is completely positive than it is doubly nonnegative, i.e. positive semidefinite and elementwise nonnegative. This necessary condition is not sufficient. If A is a nonnegative symmetric matrix and its comparison matrix is an M-matrix than A is completely positive. This sufficient condition is not necessary. In the talk I will describe qualitative conditions on the matrices (in terms of odd cycles in the associated graphs) under which the necessary condition is sufficient and the sufficient condition is necessary. I will also discuss the smallest possible number of columns of B in a factrization A=BB^T of a completely positive matrix.


Speaker's Contact Info: berman(at-sign)techunix.technion.ac.il


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