In mathematics, specifically linear algebra, a real symmetric matrix A is copositive if

for every nonnegative vector ${\displaystyle x\geq 0}$ (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.

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: $${\displaystyle {\begin{pmatrix}1&-1&1&1&-1\\-1&1&-1&1&1\\1&-1&1&-1&1\\1&1&-1&1&-1\\-1&1&1&-1&1\end{pmatrix}}}$$

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: