In mathematics, specifically linear algebra, a real symmetric matrix is copositive if
for every nonnegative vector (where the inequalities should be understood coordinate-wise). Some authors do not require to be symmetric. The collection of all copositive matrices is a proper cone; it includes as a subset the collection of real positive-definite matrices.
Copositive matrices find applications in economics, operations research, and statistics.
Examples Every real positive-semidefinite matrix is copositive by definition. Every symmetric nonnegative matrix is copositive. This includes the zero matrix. The exchange matrix is copositive but not positive-semidefinite.
Properties It is easy to see that the sum of two copositive matrices is a copositive matrix. More generally, any conical combination of copositive matrices is copositive.
Let be a copositive matrix. Then we have that every principal submatrix of is copositive as well. In particular, the entries on the main diagonal must be nonnegative. the spectral radius is an eigenvalue of .
Every copositive matrix of order less than 5 can be expressed as the sum of a positive semidefinite matrix and a nonnegative matrix. A counterexample for order 5 is given by a copositive matrix known as Horn-matrix:
Characterization The class of copositive matrices can be characterized using principal submatrices. One such characterization is due to Wilfred Kaplan: A real symmetric matrix is copositive if and only if every principal submatrix of has no eigenvector with associated eigenvalue . Several other characterizations are presented in a survey by Ikramov, including: Assume that all the off-diagonal entries of a real symmetric matrix A are nonpositive. Then A is copositive if and only if it is positive semidefinite.
The problem of deciding whether a matrix is copositive is co-NP-complete.
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