Completely Positive Matrix

A completely positive matrix is a real n×n square matrix A=(a_(ij)) that can be factorized as


where B^(T) stands for the transpose of B and B is any (not necessarily square) n×m matrix with nonnegative elements. The least possible number of columns (m) of B is called the factorization index (or the cp-rank) of A. The study of complete positivity originated in inequality theory and quadratic forms (Diananda 1962, Hall and Newman 1963).

There are two basic problems concerning complete positivity.

1. When is a given n×n real matrix A completely positive?

2. How can the cp-rank of A be calculated?

These two fundamental problems still remains open. The applications of completely positive matrices can be found in block designs (Hall and Newman 1963) and economic modelling (Gray and Wilson 1980).

See also

Copositive Matrix, Doubly Nonnegative Matrix

This entry contributed by Changqing Xu

Explore with Wolfram|Alpha


Ando, T. Completely Positive Matrices. Lecture Notes. Sapporo, Japan, 1991.Berman, A. "Complete Positivity." Linear Algebra Appl. 107, 57-63, 1988.Berman, A. "Completely Positive Graphs." In Combinatorial and Graph-Theoretical Problems in Linear Algebra: Papers from the IMA Workshop held in Minneapolis, Minnesota, November 11-15, 1991 (Ed. R. A. Brualdi, S. Friedland, and V. Klee). New York: Springer-Verlag, pp. 229-233, 1991.Berman, A. and Shaked-Monderer, N. Completely Positive Matrices. Singapore: World Scientific, 2003.Diananda, P. H. "On Nonnegative Forms in Real Variables Some or All of Which Are Nonnegative." Proc. Cambridge Philos. Soc. 58, 17-25, 1962.Gray, L. J. and Wilson, D. G. "Nonnegative Factorization of Positive Semidefinite Nonnegative Matrices." Linear Algebra Appl. Appl. 31, 119-127, 1980.Hall, M. Jr. and Newman, M. "Copositive and Completely Positive Quadratic Forms." Proc. Cambridge Philos. Soc. 59, 329-339, 1963.

Referenced on Wolfram|Alpha

Completely Positive Matrix

Cite this as:

Xu, Changqing. "Completely Positive Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications