Completely positive matrices associated with<i>M</i>-matrices

Drew, John H.; Johnson, Charles R.; Loewy, Raphael

doi:10.1080/03081089408818334

Cited by 61 publications

(50 citation statements)

References 11 publications

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“…For n = 5 we have p n = n 2 /4 [20]. It was conjectured in [12] that this equality holds for all n ≥ 5, however counter examples to this conjecture for n = 7, . .…”

Section: Perron-frobenius Perturbationsmentioning

confidence: 86%

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The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Bomze

Dickinson

Still

2015

Linear Algebra and its Applications

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We study the topological properties of the cp-rank operator cp(A) and the related cp-plus-rank operator cp + (A) (which is introduced in this paper) in the set S n of symmetric n × n-matrices. For the set of completely positive matrices, CP n , we show that for any fixed p the set of matrices A satisfying cp(A) = cp + (A) = p is open in S n \ bd (CP n ). We also prove that the set A n of matrices with cp(A) = cp + (A) is dense in S n . By applying the theory of semi-algebraic sets we are able to show that membership in A n is even a generic property. We furthermore answer several questions on the existence of matrices satisfying cp(A) = cp + (A) or cp(A) = cp + (A), and establish genericity of having infinitely many minimal cp-decompositions.

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“…For n = 5 we have p n = n 2 /4 [20]. It was conjectured in [12] that this equality holds for all n ≥ 5, however counter examples to this conjecture for n = 7, . .…”

Section: Perron-frobenius Perturbationsmentioning

confidence: 86%

“…4.1, Thm. 4.1].Hence it remains to prove(12). Similarly to before, from Lemma 2.2, for all k ∈ {1, .…”

mentioning

confidence: 86%

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Bomze

Dickinson

Still

2015

Linear Algebra and its Applications

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“…The second theorem we use is obtained by combining a result of [11] with a result of [5], and the above observation. Theorem 1.6.…”

Section: For Every Cp Matrix a Cp-rank A ≥ C(g(a))mentioning

confidence: 99%

“…Theorem 1.6. [11,5] If G is a triangle free graph on n ≥ 4 vertices and A is a CP matrix realization of G, then…”

Section: For Every Cp Matrix a Cp-rank A ≥ C(g(a))mentioning

confidence: 99%

Minimal CP rank

Shaked-Monderer¹

2001

ELA

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“…A major line of research in the linear algebra literature is concerned with determining the minimal number k of factors necessary in such a decomposition. This quantity is called the cp-rank, and is conjectured [25] to be n 2 /4 if n is the order of the matrix. See [6] for more details on the cp-rank.…”

Section: Complete Positivitymentioning

confidence: 99%

Copositive Programming – a Survey

Dür

2010

Recent Advances in Optimization and Its Applications in Engineering

179

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Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.

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Completely positive matrices associated withM-matrices

Cited by 61 publications

References 11 publications

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Minimal CP rank

Copositive Programming – a Survey

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