A doubly nonnegative matrix is a real positive semidefinite 
square matrix with
nonnegative entries. Any doubly nonnegative matrix 
of order 
can be expressed as a Gram matrix
of 
vectors 
(where 
is the rank of 
),
with each pair of vectors possessing a nonnegative inner
product, i.e., 
. Every completely
positive matrix is doubly nonnegative.
Doubly Nonnegative Matrix
See also
Completely Positive Matrix, Copositive MatrixThis 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.Referenced on Wolfram|Alpha
Doubly Nonnegative MatrixCite this as:
Xu, Changqing. "Doubly Nonnegative Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DoublyNonnegativeMatrix.html