#### Title

Modern Theory of Copositive Matrices

5-2022

#### Access Type

Honors Thesis -- Open Access

#### Degree Name

Bachelors of Science (BS)

#### Department

Mathematics

Charles R. Johnson

Pierre Clare

Huajie Shao

#### Abstract

Copositivity is a generalization of positive semidefiniteness. It has applications in theoretical economics, operations research, and statistics. An \$n\$-by-\$n\$ real, symmetric matrix \$A\$ is copositive (CoP) if \$x^T Ax \ge 0\$ for any nonnegative vector \$x \ge 0.\$ The set of all CoP matrices forms a convex cone. A CoP matrix is ordinary if it can be written as the sum of a positive semidefinite (PSD) matrix and a symmetric nonnegative (sN) matrix. When \$n < 5,\$ all CoP matrices are ordinary. However, recognizing whether a given CoP matrix is ordinary and determining an ordinary decomposition (PSD + sN) is still an unsolved problem. Here, we give an overview on modern theory of CoP matrices, talk about our progress on the ordinary recognition and decomposition problem, and emphasis the graph theory aspect of ordinary CoP matrices.

COinS