Linear Algebra and Its Applications, ISSN 0024-3795, 2009, Volume 431, Issue 9, pp. 1539 - 1552

The convex cone of n × n completely positive (CP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including...

Copositive matrices | Doubly nonnegative matrices | Completely positive matrices

Copositive matrices | Doubly nonnegative matrices | Completely positive matrices

Journal Article

SIAM Review, ISSN 0036-1445, 12/2010, Volume 52, Issue 4, pp. 593 - 629

This work surveys essential properties of the so-called copositive matrices, the study of which has been spread over more than fifty-five years. Special...

Algorithms | Spectral theory | Linear algebra | Eigenvalues | SURVEY and REVIEW | Matrices | Mathematical inequalities | Mathematical vectors | Mathematical functions | Polynomials | Copositive matrix | Copositive programming | Pareto eigenvalues | Quadratic programming | Convex cone | Copositivity test | MATHEMATICS, APPLIED | CONVEX CONES | CRITICAL ANGLES | EIGENVALUE COMPLEMENTARITY-PROBLEM | QUADRATIC OPTIMIZATION PROBLEMS | quadratic programming | OPTIMALITY CRITERIA | SEMIDEFINITE | FORMS | copositive programming | copositive matrix | convex cone | LOCAL MINIMA | INEQUALITY CONSTRAINTS | copositivity test | POSITIVE MATRICES | Usage | Hilbert space | Analysis | Convex analysis | Matrix | Mathematics

Algorithms | Spectral theory | Linear algebra | Eigenvalues | SURVEY and REVIEW | Matrices | Mathematical inequalities | Mathematical vectors | Mathematical functions | Polynomials | Copositive matrix | Copositive programming | Pareto eigenvalues | Quadratic programming | Convex cone | Copositivity test | MATHEMATICS, APPLIED | CONVEX CONES | CRITICAL ANGLES | EIGENVALUE COMPLEMENTARITY-PROBLEM | QUADRATIC OPTIMIZATION PROBLEMS | quadratic programming | OPTIMALITY CRITERIA | SEMIDEFINITE | FORMS | copositive programming | copositive matrix | convex cone | LOCAL MINIMA | INEQUALITY CONSTRAINTS | copositivity test | POSITIVE MATRICES | Usage | Hilbert space | Analysis | Convex analysis | Matrix | Mathematics

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 10/2014, Volume 459, pp. 154 - 174

Let A be an element of the copositive cone Cn. A zero u of A is a nonzero nonnegative vector such that uTAu=0. The support of u is the index set suppu⊂{1,…,n}...

Irreducibility | Extreme ray | Copositive matrix | MATHEMATICS | MATHEMATICS, APPLIED

Irreducibility | Extreme ray | Copositive matrix | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Annals of Operations Research, ISSN 0254-5330, 8/2016, Volume 243, Issue 1, pp. 375 - 382

In this article, we study the properties of some matrix classes using principal pivot transform (PPT). These matrices with some additional conditions have...

Almost strictly semimonotone matrix | Almost fully copositive matrix | Principal pivot transforms | Business and Management | Semimonotone matrix | Exact order 2 fully copositive matrix | Theory of Computation | Q_{0}$$ Q 0 -matrix | Operation Research/Decision Theory | Almost copositive matrix | Combinatorics | matrix | COPOSITIVE MATRICES | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Q-matrix | LINEAR COMPLEMENTARITY-PROBLEM | Transformations (Mathematics) | Matrices | Research | Mathematical research | Texts | Operations research | Pivots | Quadratic programming | Transforms | Linear algebra

Almost strictly semimonotone matrix | Almost fully copositive matrix | Principal pivot transforms | Business and Management | Semimonotone matrix | Exact order 2 fully copositive matrix | Theory of Computation | Q_{0}$$ Q 0 -matrix | Operation Research/Decision Theory | Almost copositive matrix | Combinatorics | matrix | COPOSITIVE MATRICES | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Q-matrix | LINEAR COMPLEMENTARITY-PROBLEM | Transformations (Mathematics) | Matrices | Research | Mathematical research | Texts | Operations research | Pivots | Quadratic programming | Transforms | Linear algebra

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 02/2017, Volume 515, pp. 53 - 86

Several results about sign properties of Metzler matrices are obtained. It is first established that checking the sign-stability of a Metzler sign-matrix can...

Positive systems | Sign-stability | Metzler matrices | LINEAR-SYSTEMS | MATHEMATICS | COPOSITIVE LYAPUNOV FUNCTIONS | MATHEMATICS, APPLIED | STABILITY | PATTERNS | INVERSE | DISCRETE-TIME-SYSTEMS | Markov processes

Positive systems | Sign-stability | Metzler matrices | LINEAR-SYSTEMS | MATHEMATICS | COPOSITIVE LYAPUNOV FUNCTIONS | MATHEMATICS, APPLIED | STABILITY | PATTERNS | INVERSE | DISCRETE-TIME-SYSTEMS | Markov processes

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 06/2017, Volume 523, pp. 46 - 51

In this note it is proved that every rational matrix which lies in the interior of the cone of completely positive matrices also has a rational...

Completely positive matrix | cp-factorization | Copositive programming | MATHEMATICS | MATHEMATICS, APPLIED | Mathematics - Optimization and Control

Completely positive matrix | cp-factorization | Copositive programming | MATHEMATICS | MATHEMATICS, APPLIED | Mathematics - Optimization and Control

Journal Article

Results in Mathematics, ISSN 1422-6383, 12/2019, Volume 74, Issue 4, pp. 1 - 10

A real symmetric quadratic form $$f = f(Z_1,\ldots ,Z_n)$$ f=f(Z1,…,Zn) in the n non-commuting indeterminates $$Z_1,\ldots ,Z_n$$ Z1,…,Zn is said to be d...

15A63 | eigenvalues | Symmetric matrix | positive form | 15A45 | test tuples | Mathematics, general | Mathematics | positive semidefinite matrix | 15B57 | copositive form | MATHEMATICS | MATHEMATICS, APPLIED

15A63 | eigenvalues | Symmetric matrix | positive form | 15A45 | test tuples | Mathematics, general | Mathematics | positive semidefinite matrix | 15B57 | copositive form | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 02/2017, Volume 514, pp. 1 - 46

Let n≥5 and let u1,…,un be nonnegative real n-vectors such that the indices of their positive elements form the sets {1,2,…,n−2},{2,3,…,n−1},…,{n,1,…,n−3},...

Zero support set | Extreme ray | Copositive matrix | POLYNOMIALS | MATHEMATICS | MATHEMATICS, APPLIED | PROGRAMS | NONNEGATIVE COEFFICIENTS | OPTIMIZATION | QUADRATIC-FORMS | Mathematics | Optimization and Control

Zero support set | Extreme ray | Copositive matrix | POLYNOMIALS | MATHEMATICS | MATHEMATICS, APPLIED | PROGRAMS | NONNEGATIVE COEFFICIENTS | OPTIMIZATION | QUADRATIC-FORMS | Mathematics | Optimization and Control

Journal Article

IEEE Transactions on Automatic Control, ISSN 0018-9286, 08/2017, Volume 62, Issue 8, pp. 4167 - 4172

It is known that the stability of a Metzler matrix can be characterized in a Routh-Hurwitz-like fashion based on a recursive application of scalar Schur...

Linear systems | Context | Symmetric matrices | stability theory | Stability criteria | Positive linear systems | Linear algebra | Numerical stability | COPOSITIVE LYAPUNOV FUNCTIONS | SYSTEMS | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Stability | Research

Linear systems | Context | Symmetric matrices | stability theory | Stability criteria | Positive linear systems | Linear algebra | Numerical stability | COPOSITIVE LYAPUNOV FUNCTIONS | SYSTEMS | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Stability | Research

Journal Article

Acta Mathematica Vietnamica, ISSN 0251-4184, 12/2018, Volume 43, Issue 4, pp. 629 - 639

This paper was presented as an invited talk in the 6th International Conference on Matrix Analysis and Applications, Duy Tan University, Da Nang City, Vietnam,...

15A23 | CP-rank | Mathematics, general | 15B48 | Mathematics | Completely positive matrices | Copositive optimization

15A23 | CP-rank | Mathematics, general | 15B48 | Mathematics | Completely positive matrices | Copositive optimization

Journal Article

Electronic Journal of Linear Algebra, 01/2018, Volume 34, Issue 1, pp. 28 - 34

Journal Article

ELECTRONIC JOURNAL OF LINEAR ALGEBRA, ISSN 1537-9582, 02/2018, Volume 34

Let A is an element of C-n be an exceptional extremal copositive n x n matrix with positive diagonal. A zero u of A is a non-zero nonnegative vector such that...

FORMS | MATHEMATICS | Extreme ray | Zero support set | Copositive matrix | CP-RANK

FORMS | MATHEMATICS | Extreme ray | Zero support set | Copositive matrix | CP-RANK

Journal Article

Networks, ISSN 0028-3045, 07/2018, Volume 72, Issue 1, pp. 128 - 150

Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone generated by the...

semidefinite programming | flows in graphs | copositive programming | approximation hierarchies | length‐bounded flows | quadratic shortest path | length-bounded flows | COMPUTATIONAL-COMPLEXITY | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NETWORKS | Hierarchies | Approximation | Mathematical analysis | Shortest-path problems | Representations | Matrix methods | Optimization

semidefinite programming | flows in graphs | copositive programming | approximation hierarchies | length‐bounded flows | quadratic shortest path | length-bounded flows | COMPUTATIONAL-COMPLEXITY | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NETWORKS | Hierarchies | Approximation | Mathematical analysis | Shortest-path problems | Representations | Matrix methods | Optimization

Journal Article

Linear and Multilinear Algebra, ISSN 0308-1087, 11/2018, Volume 66, Issue 11, pp. 2151 - 2155

Let be an extremal copositive matrix with unit diagonal. Then the minimal zeros of A all have supports of cardinality two if and only if the elements of A are...

Copositive matrix | extreme ray | 15A48 | minimal zero | 15A21 | MATHEMATICS | CP-RANK | QUADRATIC-FORMS | Mathematics | Optimization and Control

Copositive matrix | extreme ray | 15A48 | minimal zero | 15A21 | MATHEMATICS | CP-RANK | QUADRATIC-FORMS | Mathematics | Optimization and Control

Journal Article

SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, 2014, Volume 35, Issue 1, pp. 127 - 142

A symmetric matrix C is completely positive (CP) if there exists an entrywise nonnegative matrix B such that C = BBT. The CP-completion problem is to study...

E-truncated ε-moment problem | Semidefinite program | E-matrices | Completely positive matrices | Matrix completion | MATHEMATICS, APPLIED | matrix completion | CONES | COPOSITIVE OPTIMIZATION | semidefinite program | epsilon-truncated Delta-moment problem | K-MOMENT PROBLEMS | EUCLIDEAN DISTANCE MATRIX | POSITIVE SEMIDEFINITE | completely positive matrices

E-truncated ε-moment problem | Semidefinite program | E-matrices | Completely positive matrices | Matrix completion | MATHEMATICS, APPLIED | matrix completion | CONES | COPOSITIVE OPTIMIZATION | semidefinite program | epsilon-truncated Delta-moment problem | K-MOMENT PROBLEMS | EUCLIDEAN DISTANCE MATRIX | POSITIVE SEMIDEFINITE | completely positive matrices

Journal Article

Linear and Multilinear Algebra, ISSN 0308-1087, 02/2015, Volume 63, Issue 2, pp. 384 - 396

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard...

copositive optimization | 90C25 | 15A23 | nonnegative factorization | 15B48 | cp-rank | completely positive matrices | FORMS | MATHEMATICS | INTERIOR | POSITIVE MATRICES | CONE | Economic impact | Approximation | Matrices (mathematics) | Mathematical analysis | Scalars | Boundaries | Matrix methods | Standards | Optimization | Mathematics - Optimization and Control

copositive optimization | 90C25 | 15A23 | nonnegative factorization | 15B48 | cp-rank | completely positive matrices | FORMS | MATHEMATICS | INTERIOR | POSITIVE MATRICES | CONE | Economic impact | Approximation | Matrices (mathematics) | Mathematical analysis | Scalars | Boundaries | Matrix methods | Standards | Optimization | Mathematics - Optimization and Control

Journal Article

SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, 2013, Volume 34, Issue 3, pp. 1384 - 1400

Among hollow, symmetric n-by-n nonnegative matrices, it is shown that any number k, 2 <= k <= n - 1 of nonpositive eigenvalues is possible. However, as n...

Copositive matrices | Ramsey numbers | Eigenvalues perturbation | Independence number | Morishima matrices | MATHEMATICS, APPLIED | independence number | copositive matrices | eigenvalues perturbation | SIGN PATTERN

Copositive matrices | Ramsey numbers | Eigenvalues perturbation | Independence number | Morishima matrices | MATHEMATICS, APPLIED | independence number | copositive matrices | eigenvalues perturbation | SIGN PATTERN

Journal Article

Linear and Multilinear Algebra, ISSN 0308-1087, 05/2017, Volume 65, Issue 5, pp. 897 - 908

A symmetric matrix is called copositive if it satisfies the inequality whenever and strictly copositive if , whenever . The ordering of a vector here is...

Copositive matrices | 15B63 | copositive-plus matrices | 15A48 | 15B57 | Moore-Penrose inverse | 15A09 | Moore–Penrose inverse | MATHEMATICS | Inheritances | Algebra | Matrices (mathematics) | Mathematical analysis | Inequalities | Images | Complement | Inverse | Vectors (mathematics)

Copositive matrices | 15B63 | copositive-plus matrices | 15A48 | 15B57 | Moore-Penrose inverse | 15A09 | Moore–Penrose inverse | MATHEMATICS | Inheritances | Algebra | Matrices (mathematics) | Mathematical analysis | Inequalities | Images | Complement | Inverse | Vectors (mathematics)

Journal Article

19.
Full Text
On generalizations of positive subdefinite matrices and the linear complementarity problem

Linear and Multilinear Algebra, ISSN 0308-1087, 10/2018, Volume 66, Issue 10, pp. 2024 - 2035

In this paper, we introduce the notion of generalized positive subdefinite matrices of level k by generalizing the definition of generalized positive...

generalized positive subdefinite matrices of level k | row sufficient matrices | copositive matrices | generalized positive subdefinite matrices | 90C33 | Linear complementarity problem | Lemke's algorithm | Lemke’s algorithm | MATHEMATICS | generalizedpositive subdefinite matrices of level k | MONOTONICITY | CONVERGENCE | SUFFICIENT MATRICES | Scientific papers | Mathematical analysis | Matrix methods | Linear algebra

generalized positive subdefinite matrices of level k | row sufficient matrices | copositive matrices | generalized positive subdefinite matrices | 90C33 | Linear complementarity problem | Lemke's algorithm | Lemke’s algorithm | MATHEMATICS | generalizedpositive subdefinite matrices of level k | MONOTONICITY | CONVERGENCE | SUFFICIENT MATRICES | Scientific papers | Mathematical analysis | Matrix methods | Linear algebra

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 10/2014, Volume 459, pp. 208 - 221

We study n×n completely positive matrices M on the boundary of the completely positive cone, namely those orthogonal to a copositive matrix S which generates a...

Nonnegative factorization | Cp-rank | Circular symmetry | Copositive optimization | Completely positive matrices | MATHEMATICS, APPLIED

Nonnegative factorization | Cp-rank | Circular symmetry | Copositive optimization | Completely positive matrices | MATHEMATICS, APPLIED

Journal Article

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