matrix sequences associated with the ducci map and the mediant construction of the rationals abstract the ducci map on r n takes a vector x 1 x n t to the vector of adjacent entry differences x 1 x 2 x n x 1 t this paper considers the case where n 3 with the focus being on those sequences of 3 3 matrices that can be used to carry out iterations of the ducci map this linear algebraic perspective reveals an unexpected connection between the ducci map the process of forming mediants of rational numbers and the stern brocot tree

a connection between schur multiplication and fourier interpolation given m n matrices a ajk and b bjk their schur product is the m n matrix a b ajkbjk for any matrix a define a s max x 1 a x where denotes the usual norm for any y a b c r3 define by f l1 f 1 a f 0 bandf 1 c and set ay abbc our main result is that for any y r3 ay s inff by f 1

triangular truncation and finding the norm of a hadamard multiplier for b a fixed matrix we study the problem of using a criterion of haagerup to find the norm of the map a a b where is the hadamard or entrywise product of matrices the techniques developed are applied to triangular truncation and it is proved that if kn is the norm of triangular truncation of n n matrices then knlogn 1

feasibility of interference alignment for the mimo interference channel we study vector space interference alignment for the multiple input multiple output interference channel with no time or frequency diversity and no symbol extensions we prove both necessary and sufficient conditions for alignment in particular we characterize the feasibility of alignment for the symmetric three user channel where all users transmit along d dimensions all transmitters have m antennas and all receivers have n antennas as well as feasibility of alignment for the fully symmetric m n channel with an arbitrary number of users an implication of our results is that the total degrees of freedom available in a k user interference channel using only spatial diversity from the multiple antennas is at most 2 this is in sharp contrast to the k 2 degrees of freedom shown to be possible by cadambe and jafar with arbitrarily large time or frequency diversity moving beyond the question of feasibility we additionally discuss computation of the number of solutions using schubert calculus in cases where there are a finite number of solutions

an operator inequality and self adjointness abstract given bounded positive invertible operators a and b on a hilbert space h it is shown that the inequality axa 1 b 1 xb 2 x holds for all bounded operators x of rank 1 if and only if b f a for some increasing function f satisfying a certain simple inequality which in the case when the spectrum of a is connected implies that b is a scalar multiple of a as an application some consequences of the corach porta recht type inequality in operator ideals are studied

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

analyzing specific cases of an operator inequality abstract let h and k be hilbert spaces and suppose a e b h and b e b k are self adjoint operators with dist a b 0 in 1983 bhatia davis and mcintosh showed that for any q e b k h we must have 2 aq qb q in this paper we specialize their inequality to the case where a q and b are 2 2 or 3 3 matrices and give sharp estimates doing so we illustrate one way that bounds on the norm of the schur product of two matrices have applications to perturbation theory by specializing the fourier transform used in the proof of the theorem above we also obtain sharp estimates in two fourier interpolation problems

some properties of circulant matrices with ducci sequences abstract a ducci sequence is the sequence x d x d 2 x generated by n tuples x x 1 x 2 x n z n where d x d x 1 x 2 x n x 2 x 1 x 3 x 2 x n x 1 equivalently the ducci sequence of n vector x may be defined as d k x where k 0 1 2 and d k x denotes k times iterated vector x in this study we examine properties of the matrix d c n which results from applying the ducci map to the rows of the circulant matrix c n with first row c 0 c 1 c n 1

maximal spectral distance abstract the exact upper bound for the norm distance between two normal matrices is given as a function of their eigenvalues among the applications we obtain an inequality related to the cartesian decomposition of a matrix and the evaluation of the hadamard multiplier norms for matrices in a certain broad class

on the norms of circulant matrices with the fibonacci and lucas numbers let us define a a i j and b b i j as nxn matrices where a i j f m o d j i n and b i j l m o d j i n we find out some bounds for the a and b matrices concerned with the spectral and euclidean norms

on the spectral norms of circulant matrices with classical fibonacci and lucas numbers entries abstract this paper is an improving of the work s solak on the norms of circulant matrices with the fibonacci and lucas numbers appl math comp 160 2005 125 132 in which the lower and upper bounds for the spectral norms of the matrices a f mod j i n i j 1 n and b l mod j i n i j 1 n are established in this new paper we compute the spectral norms of these matrices

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

the singular values of a b and a ib we obtain several inequalities relating the singular values of a b and those of a ib when a and b are hermitian and when one or both of them are positive semidefinite

comparison of h and state space induced quotient topologies on stable minimal systems it is shown that on the set of w input p output asymptotically stable minimal nth order systems that the h p m and the rn quoiient induced topologies coincide implications of this in frequency domain system identification are discussed

adaptive estimation in time series regression models this work develops adaptive estimators for a linear regression model with serially correlated errors we show that these results continue to hold when the order of the arma process characterizing the errors is unknown the finite sample results are promising indicating that substantial efficiency gains may be possible for samples as small as 50 observations we use these estimators to investigate the behavior of the forward foreign exchange market

on the determinants and inverses of circulant matrices with fibonacci and lucas numbers abstract let a n circ f 1 f 2 f n and b n circ l 1 l 2 l n be circulant matrices where f n is the fibonacci number and l n is the lucas number we prove that a n is invertible for n 2 and b n is invertible for any positive integer n afterwards the values of the determinants of matrices a n and b n can be expressed by utilizing only the fibonacci and lucas numbers in addition the inverses of matrices a n and b n are derived

the quadratic irrationals and ducci matrix sequences abstract the ducci map is defined by taking a vector v 1 v n t r n to v 1 v 2 v n v 1 t we concern ourselves with the ducci map s action in r 3 establishing a connection between the sequences of matrices associated with the action of the ducci map continued fraction representations of the real numbers and the stern brocot tree it is shown that the real numbers have a representation via sequences of ducci matrices and in this ducci number system there are essentially three types of matrix sequences one type corresponding to the rationals one to the quadratic irrationals and then those types that correspond to all other real numbers this mirrors the situation for continued fraction representations of the reals it follows that the ducci map on r 3 is closely connected to the euclidean algorithm and through its action locates best rational approximations to the irrationals

the properties of the parameterization of armax systems and their relevance for structural estimation and dynamic specification armax systems in structural specification are considered topological and geometric properties of the parameter space and of the parameterization which are important for the properties of the estimators regardless of their special form are investigated and the specification of the maximum lag lengths is discussed

schur norms of bicirculant matrices abstract a formula is given for computing schur norm of a block circulant matrix with circulant blocks together with some generalizations

spectral decomposition of real circulant matrices abstract this paper presents spectral decompositions i e eigendecompositions and singular value decompositions of four types of real circulant matrices right and left circulants whose elements topple from right to left or from left to right respectively as well as skew right and skew left circulants whose elements change their sign when toppling are analyzed the inherent periodicity of circulant matrices means that they are closely related to fourier analysis and group theory this relationship is utilized in the spectral decompositions of this paper

optimal measures for toeplitz matrices abstract we extend and refine a result of r mceachin concerning 2 2 toeplitz matrices even in the complex case the schur norm of such a matrix is equal to the total variation of an appropriately chosen representing measure the measure may be supported on two or three points from this phenomenon we derive formulas for the schur norms of the matrices we point out a connection with the extreme points of the unit ball in a certain function space

model sets and parametrizations for identification of multivariable equation error models abstract equation error or linear regression models are known to inherently require the a priori choice for specific signal variables to be considered as regressand and or regressor this implies that a model set should be a priori restricted in some way in order to define an acceptable identification problem in the case of approximate identification i e the system to be modelled is not contained in the model set this restriction acts as a design variable with the identified models being dependent on its specific choice in this paper the necessity of this restriction is quantified by the property of discriminability i e the ability of an identification criterion to distinguish between all the different models in a model set employing a deterministic signal oriented framework several sets of sufficient conditions are derived for model sets to be discriminable by a least squares identification criterion to this end use is made of polynomial model representations in two shift operators although it is of a different nature the problem discussed is shown to be closely related to the problem of constructing identifiable parametrizations for sets of rational transfer functions it is shown that the pseudo canonical or overlapping parametrization of all transfer functions with fixed mcmillan degree constitutes a nonoverlapping set of equation error models that is discriminable by a least squares identification criterion

general results on the mcmillan degree and the kronecker indices of arma and mfd models this paper shows that the mcmillan degree of general arma and mfd models is equal to the pole zero excess of the matrix consisting of the polynomial factors furthermore the left kronecker indices are equal to the row degrees of this matrix if and only if it is row reduced and irreducible for left coprime arma and mfd models the mcmillan degree and the left kronecker indices are related to the determinantal degree and the row degrees of a suitable submatrix of the polynomial factors under certain necessary and sufficient conditions this information can even be inferred from the denominator matrices in the arma and mfd models finally a rank test is presented for actually computing the mcmillan degree of left coprime arma and mfd models

finding norms of hadamard multipliers abstract the norm of a matrix b as a hadamard multiplier is the norm of the map x at x b where is the hadamard or entrywise product of matrices watson proposed an algorithm for finding lower bounds for the hadamard multiplier norm of a matrix it is shown how watson s algorithm can be used to give upper bounds as well which in many cases yield the hadamard multiplier norm to any desired accuracy a sharp form of wittstock s decomposition theorem is proved for the special case of hadamard multiplication

index of hadamard multiplication by positive matrices ii for each n n positive semidefinite matrix a we define the minimal index i a max 0 a b b for all b 0 and for each norm n the n index in a min n a b b 0 and n b 1 where a b aijbij is the hadamard or schur product of a aij and b bij and b 0 means that b is a positive semidefinite matrix a comparison between these indexes is done for different choices of the norm n as an application we find for each bounded invertible selfadjoint operator s on a hilbert space the best constant m s such that sts s 1ts 1 m s t for all t 0

extreme points and optimal measures abstract we provide a new and more efficient proof of our earlier result that every 2 2 toeplitz matrix m has a representing measure that is optimal in the sense that m s the norm of m as a schur multiplier this result is seen to follow from some elementary observations about extreme points in the unit ball of trigonometric trinomials we also discuss the complete characterization of such extreme points

an application of the grobner basis in computation for the minimal polynomials and inverses of block circulant matrices abstract algorithms for the minimal polynomial and the inverse of a level n r1 r2 rn block circulant matrix over any field are presented by means of the algorithm for the grobner basis for the ideal of the polynomial ring over the field and two algorithms for the inverse of a level n r1 r2 rn block circulant matrix over a quaternion division algebra are given which can be realized by cocoa 4 0 an algebraic system over the field of rational numbers or the field of residue classes of modulo a prime number

on some operator inequalities abstract for hilbert space operators s t with s invertible and self adjoint corach porta and recht recently proved that sts 1 s 1 ts 2 t a generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic geometric mean inequality some related inequalities are also discussed

on the degrees of freedom achievable through interference alignment in a mimo interference channel consider a k user flat fading mimo interference channel where the kth transmitter or receiver is equipped with mk respectively nk antennas if an exponential in k number of generic channel extensions are used either across time or frequency cadambe and jafar 1 showed that the total achievable degrees of freedom dof can be maximized via interference alignment resulting in a total dof that grows linearly with a even if mk and nk are bounded in this work we consider the case where no channel extension is allowed and establish a general condition that must be satisfied by any degrees of freedom tuple d1 d2 dk achievable through linear interference alignment for a symmetric system with mk m nk n dk d for all k this condition implies that the total achievable dof cannot grow linearly with k and is in fact no more than k m n k 1 we also show that this bound is tight when the number of antennas at each transceiver is divisible by d the number of data streams per user

properties of the parametrization of monic arma systems abstract some properties of the parametrization of multi input multi output monic arma systems with prescribed column degrees are considered in particular we give a minimal cover for the set of transfer functions corresponding to all arma systems in terms of this parametrization we investigate the relation between the kronecker indices and the prescribed column degrees and we investigate the boundary of a certain parameter space and of the corresponding set of transfer functions

a feasibility test for linear interference alignment in mimo channels with constant coefficients in this paper we consider the feasibility of linear interference alignment ia for multiple input multiple output mimo channels with constant coefficients for any number of users antennas and streams per user and propose a polynomial time test for this problem combining algebraic geometry techniques with differential topology ones we first prove a result that generalizes those previously published on this topic in particular we consider the input set complex projective space of mimo interference channels the output set precoder and decoder grassmannians and the solution set channels decoders and precoders satisfying the ia polynomial equations not only as algebraic sets but also as smooth compact manifolds using this mathematical framework we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective if that mapping is not surjective then the solution variety projects into the input space in a singular way and the projection is a zero measure set this result naturally yields a simple feasibility test which amounts to checking the rank of a matrix we also provide an exact arithmetic version of the test which proves that testing the feasibility of ia for generic mimo channels belongs to the bounded error probabilistic polynomial complexity class

the feasibility conditions for interference alignment in mimo networks interference alignment ia has attracted great attention in the last few years for its breakthrough performance in interference networks however despite the numerous works dedicated to ia the feasibility conditions of ia remains unclear for most network topologies the ia feasibility analysis is challenging as the ia constraints are sets of high degree polynomials for which no systematic tool to analyze the solvability conditions exists in this work by developing a new mathematical framework that maps the solvability of sets of polynomial equations to the linear independence of their first order terms we propose a sufficient condition that applies to mimo interference networks with general configurations we have further proved that this sufficient condition coincides with the necessary conditions under a wide range of configurations these results further consolidate the theoretical basis of ia

local parameterizations of armax systems with nonlinear restrictions abstract a necessary and sufficient condition of local identifiability is derived by using the well known rank theorem using this condition a suitable local parametrization is defined the topological and geometrical properties of this parametrization are investigated the results obtained generalize the corresponding results of deistler and wang 1989 which deal with essentially the systems with linear restrictions the treatment covers also the systems without a priori restrictions

geometry and the norms of hadamard multipliers abstract the norm of a matrix b as a hadamard multiplier is the norm of the map x x b where is the hadamard or entrywise product of matrices we give a geometric interpretation to a criterion of haagerup and use this interpretation to find an explicit formula for the hadamard multiplier norms of real 2 2 matrices for hermitian matrices with one positive eigenvalue we give necessary and sufficient conditions for the norm to be the magnitude of the largest entry

feasibility of single beam interference alignment in multi carrier interference channels sun and luo recently showed that if the vector space single beam interference alignment problem for a k user l carrier interference channel is feasible then k leq 2l 2 we prove the converse that if k leq 2l 2 then the problem is feasible i e that the requisite beamformers do exist

a note on the arithmetic geometric mean inequality for matrices abstract we give a simple proof of the inequality aa x xbb 2 a ab where a b and x are arbitrary n n matrices and is any unitarily invariant norm for the schatten p norms p with 1 p aa x xbb p 2 a xb p if and only if aa x xbb

the common structure of parametrizations for linear systems abstract a general framework for parametrizations of armax systems in the case of essentially affine restrictions is described for this general case topological and geometrical properties of the parametrizations are derived in a unified way our treatment includes for instance echelon canonical forms the overlapping description of the manifold of all transfer functions of order n and structural identifiability by affine restrictions also the case of polynomial zero restrictions is considered