Non-commutative partial matrix convexity

Hay, Damon M.; Helton, J. William; Lim, Adrian P. C.; McCullough, Scott

doi:10.1512/iumj.2008.57.3638

Cited by 16 publications

(15 citation statements)

References 8 publications

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“…If, on the other hand, p 2d (X) is never positive definite, then det(p 2d (X)) = 0 for all n and X ∈ S n (R g ). An application of the Guralnick-Small lemma as found in [8] then gives the contradiction that p 2d is the zero polynomial. Thus there is an n and an X ∈ S n (R g ) such that p 2d (X) ≻ 0.…”

Section: The Sum Of Squares Casementioning

confidence: 98%

“…Namely there is just one a variable and q(a, x) = a−p(x) for a polynomial p in the variables x alone. For comparison, a main result of [8] says, generally, if q(A, x) is convex in x for each fixed A, then q(a, x) = L(a, x) + h j (a, x) * h j (a, x) where L has degree at most one in x and each h j is linear in x. The articles [3,4,5,9] contain results for polynomials f whose positivity set -namely the set of those X such that f (X) ≻ 0 -is convex.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-convex free polynomials

Balasubramanian

McCullough

2014

Proc. Amer. Math. Soc.

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Let R ⟨ x ⟩ \mathbb R\langle x \rangle denote the ring of polynomials in g g freely noncommuting variables x = ( x 1 , … , x g ) x=(x_1,\dots ,x_g) . There is a natural involution ∗ * on R ⟨ x ⟩ \mathbb R\langle x \rangle determined by x j ∗ = x j x_j^*=x_j and ( p q ) ∗ = q ∗ p ∗ (pq)^*=q^* p^* , and a free polynomial p ∈ R ⟨ x ⟩ p\in \mathbb R\langle x \rangle is symmetric if it is invariant under this involution. If X = ( X 1 , … , X g ) X=(X_1,\dots ,X_g) is a g g tuple of symmetric n × n n\times n matrices, then the evaluation p ( X ) p(X) is naturally defined and further p ∗ ( X ) = p ( X ) ∗ p^*(X)=p(X)^* . In particular, if p p is symmetric, then p ( X ) ∗ = p ( X ) p(X)^*=p(X) . The main result of this article says if p p is symmetric, p ( 0 ) = 0 p(0)=0 and for each n n and each symmetric positive definite n × n n\times n matrix A A the set { X : A − p ( X ) ≻ 0 } \{X:A-p(X)\succ 0\} is convex, then p p has degree at most two and is itself convex, or − p -p is a hermitian sum of squares.

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Section: The Sum Of Squares Casementioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Quasi-convex free polynomials

Balasubramanian

McCullough

2014

Proc. Amer. Math. Soc.

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“…Since having an LMI is seemingly more restrictive than convexity, there has been the hope, indeed expectation, that some practical class of convex situations has been missed. The problem solved here (though not operating at full engineering generality, see [HHLM08]) is a paradigm for the type of algebra occurring in systems problems governed by signal-flow diagrams; such physical problems directly present non-commutative semi-algebraic sets. Theorem 3.3 gives compelling evidence that all such convex situations are associated to some LMI.…”

Section: Theorem 34 Suppose P Satisfies the Conditions Of Assumptiomentioning

confidence: 99%

Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

Helton

Klep

McCullough

2011

International Series in Operations Research &Amp; Management Science

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Abstract. One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are called "dimension-free". Analyzing dimension-free problems has led to the development recently of a non-commutative (nc) real algebraic geometry (RAG) which, when combined with convexity, produces dimension-free Semidefinite Programming. This article surveys what is known about convexity in the non-commutative setting and nc SDP and includes a brief survey of nc RAG. Typically, the qualitative properties of the non-commutative case are much cleaner than those of their scalar counterpartsvariables in R g . Indeed we describe how relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure.

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“…Helton and McCullough in [HM04] showed that if a noncommuting polynomial p(x) in noncommuting (free) variables is matrix convex, then p has degree two or less. In [HHLM08], Hay, Helton, Lim, and McCullough showed that a noncommuting polynomial p(a, x) in two classes of noncommuting variables that is matrix convex (in x) has degree two or less. The main result of this paper, Theorem 1.4, implies this conclusion for classes of convex noncommuting functions far more general than polynomials.…”

Section: Introductionmentioning

confidence: 99%