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Copositive Matrix

A copositive matrix is a real n×n square matrix A=(a_(ij)) that makes the corresponding quadratic form

f(x)=x^(T)Ax

nonnegative for all nonnegative n-vectors x. Copositive matrices have many applications including in the field of control theory.

The cone consisting of all copositive matrices of order n is just the dual cone of all completely positive matrices of order n.

See also

Completely Positive Matrix, Doubly Nonnegative Matrix

This entry contributed by Changqing Xu

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References

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.

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Copositive Matrix

Cite this as:

Xu, Changqing. "Copositive Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CopositiveMatrix.html

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