arXiv subjects

Igor Klep

Publications and source records attributed to Igor Klep.

88 records · Page 5Linked to original sources

arXiv2012

The matricial relaxation of a linear matrix inequality

Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask: (Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables x_j. With this relaxation, the domination questions (Q1) and (Q2) have elegant answers, indeed reduce to constructible semidefinite programs. Assume there is an X such that L_1(X) and L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^* L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are unitarily equivalent. Such algebraic certificates are typically called Positivstellensaetze and the above are examples of such for linear polynomials. The paper goes on to derive a cleaner and more powerful Putinar-type Positivstellensatz for polynomials positive on a bounded set of the form {X | L(X) PsD}. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map from a subspace of matrices to a matrix algebra is "completely positive".

math.OA↗

arXiv2012

The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution

In 1976 Procesi and Schacher developed an Artin-Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.

math.RA↗

arXiv2011

Semidefinite programming in matrix unknowns which are dimension free

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 counterparts - variables in R^g. Indeed we describe how relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure.

math.OA↗

arXiv2011

Infeasibility certificates for linear matrix inequalities

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality is infeasible if and only if -1 lies in the quadratic module associated to it. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certificate, we use a more involved algebraic certificate motivated by the real radical and Prestel's theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination.

math.OC↗

arXiv2011

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.

math.RA↗

arXiv2010

Proper Analytic Free Maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D' in g' variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D'. Assuming that both domains contain 0, we show that if f:D->D' is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g', then f is invertible and f^(-1) is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D'.

math.FA↗

arXiv2010

Analytic mappings between noncommutative pencil balls

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. In an earlier paper we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call "NC ball maps". In this paper we turn to a more general dimension-free ball B_L, called a "pencil ball", associated with a homogeneous linear pencil L(x):= A_1 x_1 + ... + A_m x_m, where A_j are complex matrices. For an m-tuple X of square matrices of the same size, define L(X):=\sum A_j \otimes X_j and let B_L denote the set of all such tuples X satisfying ||L(X)||<1. We study the generalization of NC ball maps to these pencil balls B_L, and call them "pencil ball maps". We show that every B_L has a minimal dimensional (in a certain sense) defining pencil L'. Up to normalization, a pencil ball map is the direct sum of L' with an NC analytic map of the pencil ball into the ball. That is, pencil ball maps are simple, in contrast to the classical result of D'Angelo on such analytic maps in C^m. To prove our main theorem, this paper uses the results of our previous paper mentioned above plus entirely different techniques, namely, those of completely contractive maps.

math.OA↗

arXiv2010

The truncated tracial moment problem

We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in non-commuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefinite and of finite rank. A truncated tracial sequence allows for such a representation if and only if one of its extensions admits a flat extension. Finally, we apply the theory via duality to investigate trace-positive polynomials in non-commuting variables.

math.FA↗

arXiv2009

Pure states, positive matrix polynomials and sums of hermitian squares

Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all real n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and the cartesian product of S with the real projective (t-1)-space. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for non-symmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.

math.OA↗

arXiv2009

A Note on Values of Noncommutative Polynomials

We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on an infinite dimensional Hilbert space.

math.OA↗

arXiv2009

Sign patterns for chemical reaction networks

Most differential equations found in chemical reaction networks (CRNs) have the form $dx/dt=f(x)= Sv(x)$, where $x$ lies in the nonnegative orthant, where $S$ is a real matrix (the stoichiometric matrix) and $v$ is a column vector consisting of real-valued functions having a special relationship to $S$. Our main interest will be in the Jacobian matrix, $f'(x)$, of $f(x)$, in particular in whether or not each entry $f'(x)_{ij}$ has the same sign for all $x$ in the orthant, i.e., the Jacobian respects a sign pattern. In other words species $x_j$ always acts on species $x_i$ in an inhibitory way or its action is always excitatory. In Helton, Klep, Gomez we gave necessary and sufficient conditions on the species-reaction graph naturally associated to $S$ which guarantee that the Jacobian of the associated CRN has a sign pattern. In this paper, given $S$ we give a construction which adds certain rows and columns to $S$, thereby producing a stoichiometric matrix $\widehat S$ corresponding to a new CRN with some added species and reactions. The Jacobian for this CRN based on $\hat S$ has a sign pattern. The equilibria for the $S$ and the $\hat S$ based CRN are in exact one to one correspondence with each equilibrium $e$ for the original CRN gotten from an equilibrium $\hat e$ for the new CRN by removing its added species. In our construction of a new CRN we are allowed to choose rate constants for the added reactions and if we choose them large enough the equilibrium $\hat e$ is locally asymptotically stable if and only if the equilibrium $e$ is locally asymptotically stable. Further properties of the construction are shown, such as those pertaining to conserved quantities and to how the deficiencies of the two CRNs compare.

math.DS↗

arXiv2008

Values of Noncommutative Polynomials, Lie Skew-Ideals and the Tracial Nullstellensatz

A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\times d$ matrices if and only if all of its values in the algebra of $d\times d$ matrices have zero trace.

math.RA↗

arXiv2008

Noncommutative ball maps

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call "NC ball maps". We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, "NC ball maps" are very simple, in contrast to the classical result of D'Angelo on such analytic maps over C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.

math.OA↗

arXiv2008

Sums of hermitian squares and the BMV conjecture

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.

math.OA↗

arXiv2008

Determinant Expansions of Signed Matrices and of Certain Jacobians

This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic test for determining when the Jacobian of a chemical reaction dynamics has a sign pattern.

math.RA↗

arXiv2006

Connes' embedding conjecture and sums of hermitian squares

We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever self-adjoint contraction matrices of the same size are substituted for the variables. These algebraic certificates involve sums of hermitian squares and commutators. We prove that they always exist for a similar nonnegativity condition where elements of separable II_1-factors are considered instead of matrices. Under the presence of Connes' conjecture, we derive degree bounds for the certificates.

math.OA↗