denote the Euclidean r-space, 
its nonnegative orthant, and 
the Euclidean complex r-space.2. Definitions, notations, and abbreviations
Let denote the Euclidean r-space, its nonnegative orthant, and the Euclidean complex r-space.
Definition 2.1. A real symmetric 
matrix A is copositive (abbreviated CP), if 
holds for every 
.
The abbreviation NCP will be used for "noncopositive".
Definition 2.2. A real symmetric matrix A is copositive of order k (abbreviated kCP), if every 
principal submatrix of A is CP.
Definition 2.3. A real symmetric matrix A is copositive of exact order k, if it is kCP, but not (k+1)CP.
An important special case for „copositive of exact order k" comes forward in the following definition.
Definition 2.4. A real symmetric matrix A is almost copositive if it is (r-1)CP, but not CP - provided that r > 1. A real symmetric matrix A = [a] of order 1 is almost copositive if a < 0.
The determinant of a quadratic matrix will be denoted by |A|, its adjoint by adjA. The element of the transposed matrix of adjA with coordinates (i, j) will be denoted by 
; this is the algebraic complement (otherwise co-factor) belonging to the entry 
of A. The adjoint of A = [a] is defined the matrix [1], regardless of the value of a.
When 0, or 1 will be used as vectors, that means they contain all zeros, or all ones, respectively, as their components.
We note that the concept of „copositive matrix" was first introduced by Motzkin in 1952, while „almost copositive matrix" by Väliaho in 1989.