Matrix in linear algebra
In mathematics, specifically linear algebra, a real symmetric matrix A is copositive if
formula_1
for every nonnegative vector formula_2 (where the inequalities should be understood coordinate-wise). Some authors do not require A 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.
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 A be a copositive matrix. Then we have that
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":
formula_4
Characterization.
The class of copositive matrices can be characterized using principal submatrices. One such characterization is due to Wilfred Kaplan:
Several other characterizations are presented in a survey by Ikramov, including:
The problem of deciding whether a matrix is copositive is co-NP-complete.