Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of ...Linear Matrix Inequalities and Bilinear Matrix Inequalities.
Let f(X1,…,Xn) be a nonzero multilinear noncommutative polynomial. If A is a unital algebra with a surjective inner derivation, then every element in A can be written as f(a1,…,an) for some ai∈A.
The truncated moment problem asks to characterize finite sequences of real numbers that are the moments of a positive Borel measure on Rn. Its tracial analog is obtained by integrating traces of ...symmetric matrices and is the main topic of this article. The solution of the bivariate quartic tracial moment problem with a nonsingular 7×7 moment matrix M2 whose columns are indexed by words of degree 2 was established by Burgdorf and Klep, while in our previous work we completely solved all cases with M2 of rank at most 5, split M2 of rank 6 into four possible cases according to the column relation satisfied and solved two of them. Our first main result in this article is the solution for M2 satisfying the third possible column relation, i.e., Y2=1+X2. Namely, the existence of a representing measure is equivalent to the feasibility problem of certain linear matrix inequalities. The second main result is a thorough analysis of the atoms in the measure for M2 satisfying Y2=1, the most demanding column relation. We prove that size 3 atoms are not needed in the representing measure, a fact proved to be true in all other cases. The third main result extends the solution for M2 of rank 5 to general Mn, n≥2, with two quadratic column relations. The main technique is the reduction of the problem to the classical univariate truncated moment problem, an approach which applies also in the classical truncated moment problem. Finally, our last main result, which demonstrates this approach, is a simplification of the proof for the solution of the degenerate truncated hyperbolic moment problem first obtained by Curto and Fialkow.
Let p be a multilinear polynomial in several noncommuting variables, with coefficients in an algebraically closed field K of arbitrary characteristic. In this paper we classify the possible images of ...p evaluated on 3 × 3 matrices. The image is one of the following:
{0},
the set of scalar matrices,
a (Zariski-)dense subset of sl3(K), the matrices of trace 0,
a dense subset of M
3(K),
the set of 3-scalar matrices (i.e., matrices having eigenvalues (β, βε, βε
2) where ε is a cube root of 1), or
the set of scalars plus 3-scalar matrices.
2010 Mathematics Subject Classification. Primary 16R99, 15A24, 17B60; Secondary 16R30.
Key words and phrases. Noncommutative polynomial, image, multilinear, matrices.
Let A be an algebra and let f be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between A,A, the linear span of commutators in A, and spanf(A), ...the linear span of the image of f in A. In particular, we show that A,A=A implies spanf(A)=A. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if C is a commutative unital algebra over a field F of characteristic 0, A is the matrix algebra Mn(C), and the polynomial f is neither an identity nor a central polynomial of Mn(F), then every commutator in A can be written as a difference of two elements, each of which is a sum of 7788 elements from f(A) (if C=F is an algebraically closed field, then 4 elements suffice). Similar results are obtained for some other algebras, in particular for the algebra B(H) of all bounded linear operators on a Hilbert space H.
The free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces Zn(f)={X∈Mn(k)g:detf(X)=0}. The main theorem of this article shows that f is irreducible if ...and only if Zn(f) is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Arising from this is a free singularity locus Nullstellensatz for noncommutative polynomials. Apart from consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.
Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative ...rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs--xy-convexity--is also considered. A characterization of xy-convex polynomials is given. Keywords: Partial convexity, biconvexity, bilinear matrix inequality (BMI), noncommutative rational function, noncommutative polynomial, realization theory.
We provide a new hierarchy of semidefinite programming relaxations, called
NCTSSOS
, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the ...exploitation of
term sparsity
hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits
correlative sparsity
for noncommutative optimization problems by Klep et al. (MP, 2021), and is the noncommutative analogue of the TSSOS framework by Wang et al. (SIAMJO 31: 114–141, 2021, SIAMJO 31: 30–58, 2021). We also propose an extension exploiting simultaneously correlative and term sparsity, as done previously in the commutative case (Wang in CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization, 2020). Under certain conditions, we prove that the optima of the NCTSSOS hierarchy converge to the optimum of the corresponding dense semidefinite programming relaxation. We illustrate the efficiency and scalability of NCTSSOS by solving eigenvalue/trace optimization problems from the literature as well as randomly generated examples involving up to several thousand variables.
This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue ...and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822–843, 2006) and Waki et al. (SIAM J Optim 17(1):218–242, 2006). The Gelfand–Naimark–Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature.
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