Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 339 / 1 / b / Solution

For $x$ in the nonnegative orthant, the positive semidefinite matrix $P$ satisfies $x^{T}Px\ge 0$. The nonnegative matrix $N$ satisfiessince every summand is nonnegative. Adding these inequalities provesConsequently $A=P+N$ is a copositive matrix. The symmetry of $N$ follows from that of $A$ and $P$; entrywise nonnegativity alone need not imply positive semidefiniteness.

This gives the inclusion of the positive-semidefinite-plus-nonnegative cone in the copositive cone. It is only a sufficient construction; the argument does not claim that every copositive matrix has this decomposition in arbitrary dimension.