Noncommutative convexity arises from linear matrix inequalities

Helton, J. William; McCullough, Scott; Vinnikov, Victor

doi:10.1016/j.jfa.2006.03.018

Cited by 127 publications

(144 citation statements)

References 27 publications

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“…See Corollaries 2.8 and 2.9 for the proofs. To obtain size bounds needed on the dimensions of the finite-dimensional representations of <X>/I for these Nullstellensätze, we employ systems theory realizations for noncommutative rational functions; see [BR11, Chapters 1 and 2] or [BGM05,HMV06,KVV09]. Our rational functions do not admit scalar regular points in general, so we present the necessary modifications of the classical theory to handle matrix centers in Section 3.…”

Section: Theorem 12 Let I ⊆ /I Be a (Formally) Rationally Resolvablmentioning

confidence: 99%

Null- and Positivstellensätze for rationally resolvable ideals

Klep

Vinnikov

Volčič

2017

Linear Algebra and its Applications

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Abstract. Hilbert's Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C[X]. In the free algebra C the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in finite-dimensional representations of C/I. In this article Nullstellensätze for a simple but important class of ideals in the free algebra -called tentatively rationally resolvable here -are presented. An ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the X variables using noncommutative rational functions in the remaining variables. Whether such an ideal I satisfies the Nullstellensatz is intimately related to embeddability of C/I into (free) skew fields. These notions are also extended to free algebras with involution. For instance, it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two-sided ideal I generated by 1 − j X ⊺ j X j . This is then applied to free real algebraic geometry: polynomials positive semidefinite on spherical isometries are sums of Hermitian squares modulo I. Similar results are obtained for nc unitary groups.

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Section: Theorem 12 Let I ⊆ /I Be a (Formally) Rationally Resolvablmentioning

confidence: 99%

Null- and Positivstellensätze for rationally resolvable ideals

Klep

Vinnikov

Volčič

2017

Linear Algebra and its Applications

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“…The theory described in [CHSY03], leads to and validates a symbolic algorithm for determining regions of convexity of non-commutative polynomials and even of non-commutative rational functions (for non-commutative rationals see [KVV09,HMV06]) which is implemented in NCAlgebra. This example is a simple special case of the following theorem.…”

Section: Convex Polynomialsmentioning

confidence: 99%

“…For instance, [HMV06] shows that if a non-commutative rational function is convex in an open set, then it is the Schur Complement of some monic linear pencil.…”

Section: Theorem 44 ([Hel02]) Every Matrix Positive Polynomial Is Amentioning

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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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php state required to recover the precise form of a recognizable series. The awkwardness of these various system interpretations for a recognizable formal power series gives some explanation as to why system operations work out well for transfer functions of SNMLSs (see section 4) but not so well for recognizable series-a point discussed in [30].…”

Section: For Each Source Vertex S ∈ S and Range Vertex R ∈ R Of G Dementioning

confidence: 99%

Structured Noncommutative Multidimensional Linear Systems

Ball¹,

Groenewald²,

Malakorn³

2005

SIAM J. Control Optim.

143

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Abstract. We introduce a class of multidimensional linear systems with evolution along a free semigroup. The transfer function for such a system is a formal power series in noncommuting indeterminates. Standard system-theoretic properties (the operations of cascade/parallel connection and inversion, controllability, observability, Kalman decomposition, state-space similarity theorem, minimal state-space realizations, Hankel operators, realization theory) are developed for this class of systems. We also draw out the connections with the much earlier studied theory of rational and recognizable formal power series. Applications include linear-fractional models for classical discretetime systems with structured, time-varying uncertainty, dimensionless formulas in robust control, multiscale systems and automata theory, and the theory of formal languages.Key words. multidimensional linear systems, free semigroup, controllability, observability, minimality, realization, formal power series, noncommuting indeterminates AMS subject classifications. 93B10, 13F25, 47A56, 93B28 DOI. 10.1137/S03630129044437501. Introduction. This paper considers extensions of standard system-theoretic ideas for classical, discrete-time, input/state/output linear systems to the case of certain types of generalized i/s/o systems having evolution along a free semigroup (in place of evolution along the nonnegative integers, as in the classical case). One can introduce formal frequency-domain techniques and arrive at a transfer function for such a system which is a formal power series in noncommuting variables; such objects have occurred in the context of the theory of formal languages and automata theory as well as in connection with realization theory for bilinear systems in the work of Schützenberger and Fliess (see [37,38,39,20,21,22,23] and the book [15] for a good survey).We first review those aspects of the classical theory which we here generalize to the setting of systems evolving on a free semigroup; this material can be found in many books on linear system and control theory (see, e.g., [32,16]). By a classical, discrete-time, i/s/o linear system (referred to here simply as a linear system for short) we mean a system Σ of linear equations of the form

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Noncommutative convexity arises from linear matrix inequalities

Abstract: 2016-12-26T15:11:00

Cited by 127 publications

References 27 publications

Null- and Positivstellensätze for rationally resolvable ideals

Null- and Positivstellensätze for rationally resolvable ideals

Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

Structured Noncommutative Multidimensional Linear Systems

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