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A convex cone is a subset $C$ of a real vector space closed under nonnegative linear combinations: $ax+by\in C$ whenever $x,y\in C$ and $a,b\ge 0$. The positive semidefinite cone and the copositive cone are examples, ordered by inclusion through the positive-semidefinite-plus-nonnegative cone.

The sums $P+N$ of a real positive semidefinite matrix $P$ and a symmetric nonnegative matrix $N$ form a convex cone inside the copositive cone. Both terms have nonnegative quadratic forms on the nonnegative orthant. The Horn copositive matrix shows that the inclusion is strict in dimension five; the sum of squares criterion for a biquadratic form explains this cone's relation to semidefinite programming.