a quadratically convergent parallel jacobi process for diagonally dominant matrices with nondistinct eigenvalues abstract this article presents a new jacobi like eigenvalue algorithm for non hermitian almost diagonal n n matrices in each step n 2 submatrices of order 2 are diagonalized the precautions for the multiple eigenvalues are based on theorems of fan and hoffman 1954 and wilkinson 1961 the proof of the quadratic convergence generalizes our previous result for distinct eigenvalues the convergence theorem is pessimistic concerning the region of attraction to a diagonal the local information structure makes the process suitable for parallelization on a hypercube or a systolic array

convergence of the largest singular value of a polynomial in independent wigner matrices for polynomials in independent wigner matrices we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables under fourth moment hypotheses we actually prove a more general result of the form no eigenvalues outside the support of the limiting eigenvalue distribution we build on ideas of haagerup schultz thorbjornsen on the one hand and bai silverstein on the other we refine the linearization trick so as to preserve self adjointness and we develop a secondary trick bearing on the calculation of correction terms instead of poincare type inequalities we use a variety of matrix identities and lp estimates the schwinger dyson equation controls much of the analysis

on the capacity of block multiantenna channels in this paper we consider point to point multiantenna channels with certain block distributional symmetries that do not require the entries of the channel matrix to be either gaussian or independent or identically distributed a main contribution is a capacity theorem for these channels which might be regarded as a generalization of telatar s theorem 1999 which reduces the numerical optimization domain in the capacity computation with this information theoretic result and some free probability arguments we prove an asymptotic capacity theorem that in addition to reducing the optimization domain does not depend on the dimension of the channel matrix this theorem allows us to apply free probability techniques to numerically compute the asymptotic capacity of the channels under consideration these theorems provide a very efficient method for numerically approximating both the capacity and a capacity achieving input covariance matrix of certain channels

matrix coefficient realization theory of noncommutative rational functions abstract noncommutative rational functions i e elements of the universal skew field of fractions of a free algebra can be defined through evaluations of noncommutative rational expressions on tuples of matrices this interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas from free real algebraic geometry to systems and control theory if a noncommutative rational function is regular at the origin it can be described by a linear object called a realization in this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rational functions and is well adapted for studying matrix evaluations of special interest are the minimal realizations which compensate the absence of a canonical form for noncommutative rational functions the non minimality of a realization is assessed by obstruction modules associated with it they enable us to devise an efficient method for obtaining minimal realizations with them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity using these results we determine concrete size bounds for rational identity testing construct minimal symmetric realizations and prove an effective local global principle for linear dependence of noncommutative rational functions

random positive operator valued measures we introduce several notions of random positive operator valued measures povms and we prove that some of them are equivalent we then study statistical properties of the effect operators for the canonical examples starting from the limiting eigenvalue distribution we derive the large system limit for several quantities of interest in quantum information theory such as the sharpness the noise content and the probability range finally we study different compatibility criteria and we compare them for generic povms

numerical range for random matrices abstract we analyze the numerical range of high dimensional random matrices obtaining limit results and corresponding quantitative estimates in the non limit case for a large class of random matrices their numerical range is shown to converge to a disc in particular numerical range of complex ginibre matrix almost surely converges to the disk of radius 2 since the spectrum of non hermitian random matrices from the ginibre ensemble lives asymptotically in a neighborhood of the unit disk it follows that the outer belt of width 2 1 containing no eigenvalues can be seen as a quantification the non normality of the complex ginibre random matrix we also show that the numerical range of upper triangular gaussian matrices converges to the same disk of radius 2 while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to 2 e

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

on the global fluctuations of block gaussian matrices in this paper we study the global fluctuations of block gaussian matrices within the framework of second order free probability theory in order to compute the second order cauchy transform of these matrices we introduce a matricial second order conditional expectation and compute the matricial second order cauchy transform of a certain type of non commutative random variables as a by product using the linearization technique we obtain the second order cauchy transform of non commutative rational functions evaluated on selfadjoint gaussian matrices

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

random matrix theory of multi antenna communications the ricean channel the use of multi antenna arrays in wireless communications through disordered media promises huge increases in the information transmission rate it is therefore important to analyse the information capacity of such systems in realistic situations of microwave transmission where the statistics of the transmission amplitudes channel may be coloured here we present an approach that provides analytic expressions for the statistics i e the moments of the distribution of the mutual information for general gaussian channel statistics the mathematical method applies tools developed originally in the context of coherent wave propagation in disordered media such as random matrix theory and replicas although it is valid formally for large antenna numbers this approach produces extremely accurate results even for arrays with as few as two antennas we also develop a method to analytically optimize over the input signal distribution which enables us to calculate analytic capacities when the transmitter has knowledge of the statistics of the channel the emphasis of this paper is on elucidating the novel mathematical methods used we do this by analysing a specific case when the channel matrix is a complex gaussian with arbitrary mean and unit covariance which is usually called the ricean channel

some controllability results for linear viscoelastic fluids we analyze the controllability properties of systems which provide a description at first approximation of a kind of viscoelastic fluid we consider linear maxwell fluids first we establish the large time approximate finite dimensional controllability of the system with distributed or boundary controls supported by arbitrary small sets then we prove the large time exact controllability of fluids of the same kind with controls supported by suitable large sets the proofs of these results rely on classical arguments in particular the approximate controllability result is implied by appropriate unique continuation properties while exact controllability is a consequence of observability inverse inequalities we also discuss questions concerning the controllability of viscoelastic fluids and some related open problems

complete diagrammatics of the single ring theorem using diagrammatic techniques we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices the formalism allows us to map two distinct areas of free random variables hermitian positive definite operators and non normal r diagonal operators we also rederive the haagerup larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach

a quadratically convergent parallel jacobi process for diagonally dominant matrices with distinct eigenvalues abstract this paper discusses a generalization for non hermitian matrices of the jacobi eigenvalue process 1846 in each step 1 2 n pairs of nondiagonal elements are annihilated in an almost diagonal matrix with distinct eigenvalues we prove that the recursively constructed sequence of matrices converges to a diagonal matrix as in the classical jacobi method the convergence is quadratic and the process is adapted to parallel implementation on an array processor or a hypercube

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

operator valued chordal loewner chains and non commutative probability abstract we adapt the theory of chordal loewner chains to the operator valued matricial upper half plane over a c algebra a we define an a valued chordal loewner chain as a subordination chain of analytic self maps of the a valued upper half plane such that each f t is the reciprocal cauchy transform of an a valued law t such that the mean and variance of t are continuous functions of t we relate a valued loewner chains to processes with a valued free or monotone independent increments just as was done in the scalar case by bauer 8 and schleisinger 48 we show that the loewner equation t f t z d f t z v t z when interpreted in a certain distributional sense defines a bijection between lipschitz mean zero loewner chains f t and vector fields v t z of the form v t z g t z where t is a generalized a valued law based on the loewner equation we derive a combinatorial expression for the moments of t in terms of t we also construct non commutative random variables on an operator valued monotone fock space which realize the laws t finally we prove a version of the monotone central limit theorem which describes the behavior of f t as t when t has uniformly bounded support

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

on measures of nonnormality of matrices abstract this paper discusses several measures of nonnormality of matrices i e functions v cn n r plus where v a 0 iff a is normal besides measures already in the literature we introduce new ones and give comparisons between them at length some of these comparisons e g c14 c15 and c17 manifest well known phenomena of ill conditioned eigenproblems

on the s transform over a banach algebra abstract the s transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a b valued banach noncommutative probability space for an arbitrary unital complex banach algebra b also a new proof of the additivity of the r transform in this setting is given

normal matrices an update abstract a list of seventy conditions on an n n complex matrix a equivalent to its being normal published nearly ten years ago by grone johnson sa and wolkowicz has proved to be very useful hoping that in an extended form it will be even more helpful we compile here another list of about twenty conditions they either have been overlooked by the authors of the original list or have appeared during the last decade

on the asymptotic distribution of block modified random matrices we study random matrices acting on tensor product spaces which have been transformed by a linear block operation using operator valued free probability theory under some mild assumptions on the linear map acting on the blocks we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution moreover using recent results on operator valued subordination we present an algorithm that computes numerically but in full generality the limiting eigenvalue distribution of the modified matrices our analytical results cover many cases of interest in quantum information theory we unify some known results and we obtain new distributions and various generalizations

cauchy noise loss for stochastic optimization of random matrix models via free deterministic equivalents abstract for random matrix models the parameter estimation based on the traditional likelihood functions is not straightforward in particular when we have only one sample matrix we introduce a new parameter optimization method for random matrix models which works even in such a case the method is based on the spectral distribution instead of the traditional likelihood in the method the cauchy noise has an essential role because the free deterministic equivalent which is a tool in free probability theory allows us to approximate the spectral distribution perturbed by cauchy noises by a smooth and accessible density function moreover we study an asymptotic property of determination gap which has a similar role as generalization gap besides we propose a new dimensionality recovery method for the signal plus noise model and experimentally demonstrate that it recovers the rank of the signal part even if the true rank is not small it is a simultaneous rank selection and parameter estimation procedure

on the asymptotic distribution of the commutator and anticommutator of random matrices we study in the limit of infinite matrix order the normalized eigenvalue counting measures of the commutator and anticommutator of two hermitian or real symmetric matrices rotated independently one respect to another by the random unitary or orthogonal haar distributed matrix we establish the convergence with probability 1 to a limiting nonrandom measure we obtain and analyze the functional equations for the stieltjes transforms of the limiting measures

orthogonal projections of the identity spectral analysis and applications to approximate inverse preconditioning many strategies for constructing different structures of sparse approximate inverse preconditioners for large linear systems have been proposed in the literature in a more general framework this paper analyzes the theoretical effectiveness of the optimal preconditioner in the frobenius norm of a linear system over an arbitrary subspace of m n left mathbb r right for this purpose the spectral analysis of the frobenius orthogonal projections of the identity matrix onto the linear subspaces of m n left mathbb r right is performed this analysis leads to a simple general criterion the effectiveness of the optimal approximate inverse preconditioners parametrized by any vectorial structure improves at the same time as the smallest singular value or the smallest eigenvalue s modulus of the corresponding preconditioned matrices increases to 1

spectral properties of polynomials in independent wigner and deterministic matrices abstract on the one hand we prove that almost surely for large dimension there is no eigenvalue of a hermitian polynomial in independent wigner and deterministic matrices in any interval lying at some distance from the supports of a sequence of deterministic probability measures which is computed with the tools of free probability on the other hand we establish the strong asymptotic freeness of independent wigner matrices and any family of deterministic matrices with strong limiting distribution

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

structured distance to normality of tridiagonal matrices abstract in this article we study the distance d measured in the frobenius norm of a tridiagonal matrix t to the set i t of similarly structured irreducible normal matrices the matrices in the closure of i t whose distance to t is d are characterized known results in the literature for the cases in which t is real or a toeplitz matrix are recovered in addition the special case in which t is a 2 toeplitz matrix is considered

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

local laws for polynomials of wigner matrices abstract we consider general self adjoint polynomials in several independent random matrices whose entries are centered and have the same variance we show that under certain conditions the local law holds up to the optimal scale i e the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory we prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i i d entries thus establishing the optimal bulk local law for these classes of ensembles in particular we generalize a similar result of anderson for anticommutator for more general polynomials our conditions are effectively checkable numerically

analytic subordination for bi free convolution in this paper we study some analytic properties of bi free additive convolution both scalar and operator valued we show that using properties of voiculescu s subordination functions associated to free additive convolution of operator valued distributions simpler formulas for bi free convolutions can be derived we use these formulas in order to prove a result about atoms of bi free additive convolutions

holder continuity of cumulative distribution functions for noncommutative polynomials under finite free fisher information abstract this paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory more precisely we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non microstates free fisher information highlighting the special case of lipschitz conjugate variables for the first time in this generality it is shown that the analytic distributions of those evaluations have holder continuous cumulative distribution functions with an explicit holder exponent that depends only on the degree of the considered polynomial for linear polynomials we reach in the case of finite non microstates free fisher information the optimal holder exponent 2 3 and get lipschitz continuity in the case of lipschitz conjugate variables in particular our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite non microstates free entropy which partially settles a conjecture of charlesworth and shlyakhtenko 8 we further provide a very general criterion that gives for weak approximations of measures having holder continuous cumulative distribution functions explicit rates of convergence in terms of the kolmogorov distance finally we combine these results to study the asymptotic eigenvalue distributions of polynomials in gues or matrices with more general gibbs laws for gibbs laws this extends the corresponding result obtained in 21 from convergence in distribution to convergence in kolmogorov distance in the gue case we even provide explicit rates which quantify results of 23 24 in terms of the kolmogorov distance

a random matrix approach to the lack of projections in cred f2 abstract in 1982 pimsner and voiculescu computed the k 0 and k 1 groups of the reduced group c algebra c red f k of the free group f k on k generators and settled thereby a long standing conjecture c red f k has no projections except for the trivial projections 0 and 1 later simpler proofs of this conjecture were found by methods from k theory or from non commutative differential geometry in this paper we provide a new proof of the fact that c red f k is projectionless the new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup ext c red f k is not a group for k 2 by the same type of methods we also obtain that two phenomena proved by bai and silverstein for certain classes of random matrices no eigenvalues outside a small neighbourhood of the support of the limiting distribution and exact separation of eigenvalues by gaps in the limiting distribution also hold for arbitrary non commutative selfadjoint polynomials of independent gue goe or gse random matrices with matrix coefficients

applications of realizations aka linearizations to free probability abstract we show how the combination of new linearization ideas in free probability theory with the powerful realization machinery developed over the last 50 years in fields including systems engineering and automata theory allows solving the problem of determining the eigenvalue distribution or even the brown measure in the non selfadjoint case of noncommutative rational functions of random matrices when their size tends to infinity along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras e g type ii 1 von neumann algebras with a precise control of the domains of the rational expressions the paper provides sufficient background information with the intention that it should be accessible both to functional analysts and to algebraists

infinite divisibility and a non commutative boolean to free bercovici pata bijection abstract we use the theory of fully matricial or non commutative functions to investigate infinite divisibility and limit theorems in operator valued non commutative probability our main result is an operator valued analogue for the bercovici pata bijection an important tool is voiculescu s subordination property for operator valued free convolution