Copositive matrix

In mathematics, specifically linear algebra, a real symmetric matrix A is copositive if

${\displaystyle x^{\top }Ax\geq 0}$

for every nonnegative vector ${\displaystyle x\geq 0}$ (where the inequalities should be understood coordinate-wise). Some authors do not require A to be symmetric. ^[1] The collection of all copositive matrices is a proper cone; ^[2] 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 ${\displaystyle {\bigl (}{\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}{\bigr )}}$ 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 A be a copositive matrix. Then we have that

• every principal submatrix of A is copositive as well. In particular, the entries on the main diagonal must be nonnegative.
• the spectral radius ρ(A) is an eigenvalue of A. ^[3]

Every copositive matrix of order less than 5 can be expressed as the sum of a positive semidefinite matrix and a nonnegative matrix. ^[4] A counterexample for order 5 is given by a copositive matrix known as Horn-matrix: ^[5] $${\displaystyle {\begin{pmatrix}1&-1&1&1&-1\\-1&1&-1&1&1\\1&-1&1&-1&1\\1&1&-1&1&-1\\-1&1&1&-1&1\end{pmatrix}}}$$

Characterization

The class of copositive matrices can be characterized using principal submatrices. One such characterization is due to Wilfred Kaplan: ^[6]

• A real symmetric matrix A is copositive if and only if every principal submatrix B of A has no eigenvector v > 0 with associated eigenvalue λ < 0.

Several other characterizations are presented in a survey by Ikramov, ^[3] including:

• 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.

The problem of deciding whether a matrix is copositive is co-NP-complete. ^[7]

References

• Berman, Abraham; Robert J. Plemmons (1979). Nonnegative Matrices in the Mathematical Sciences . Academic Press. ISBN   0-12-092250-9.

1. Changqing Xu. "Copositive matrix". MathWorld . Retrieved 23 September 2024.
2. Copositive matrix at PlanetMath .
3. Ikramov, Kh. D.; Savel'eva, N. V. (1 January 2000). "Conditionally definite matrices" . Journal of Mathematical Sciences. 98 (1): 1–50. doi:10.1007/BF02355379. ISSN   1573-8795.
4. Diananda, P. H. (January 1962). "On non-negative forms in real variables some or all of which are non-negative" . Mathematical Proceedings of the Cambridge Philosophical Society . 58 (1): 17–25. doi:10.1017/S0305004100036185. ISSN   1469-8064.
5. Dür, Mirjam (2010). "Copositive Programming – a Survey" (PDF). In Diehl, Moritz; Glineur, Francois; Jarlebring, Elias; Michiels, Wim (eds.). Recent Advances in Optimization and its Applications in Engineering. Berlin, Heidelberg: Springer. pp. 3–20. doi:10.1007/978-3-642-12598-0_1. ISBN   978-3-642-12598-0.
6. Kaplan, Wilfred (1 July 2000). "A test for copositive matrices". Linear Algebra and Its Applications. 313 (1): 203–206. doi:10.1016/S0024-3795(00)00138-5. ISSN   0024-3795.
7. Schweighofer, Markus; Vargas, Luis Felipe (19 October 2023). "Sum-of-squares certificates for copositivity via test states". arXiv: 2310.12853 [math.AG].