Copositive matrix
In mathematics, specifically linear algebra, a real [symmetric matrix] is copositive if
for every nonnegative vector . Some authors do not require 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.
Examples
- Every real positive-semidefinite matrix is copositive by definition.
- Every symmetric nonnegative matrix is copositive. This includes the zero matrix.
- The exchange matrix is copositive but not positive-semidefinite.
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 be a copositive matrix. Then we have that
- every principal submatrix of is copositive as well. In particular, the entries on the main diagonal must be nonnegative.
- the spectral radius is an eigenvalue of.
Characterization
The class of copositive matrices can be characterized using principal submatrices. One such characterization is due to Wilfred Kaplan:- A real symmetric matrix is copositive if and only if every principal submatrix of has no eigenvector with associated eigenvalue.
- Assume that all the off-diagonal entries of a real symmetric matrix A are nonpositive. Then A is copositive if and only if it is positive semidefinite.