loewner and bezout matrices abstract the notion of intertwining matrices introduced recently by the authors is used to study loewner matrices as in the case of hankel and bezout matrices compatible with a polynomial f of degree not necessarily equal to the size of the matrices the theory requires a careful handling of polynomials with a root at infinity the authors establish first some technical results essentially of a projective character concerning divisibility of such polynomials these are then used to represent products of the form b1b 12b3b 14 b 12rb2r 1 where the bj are bezoutian matrices corresponding to a polynomial f again in the form of a bezoutian matrix corresponding to the same f this result can be deduced from the barnett formula if the degree of f equals the size of the matrices but requires the intertwining theory if this assumption is dropped among other results a strengthening and a conceptual proof of a recent result of z vavrin on inverses of loewner matrices is given in the last section the authors investigate analogies of barnett type formulae for bezoutian matrices b f g where the degree of f is allowed to be smaller than the size of the matrix the barnett formula does not extend immediately but it can be shown that an analogous formula holds asymptotically in the general case

a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

asymmetricalgebraicriccatiequation ahomeomorphicparametrizationofthesetof solutions abstract in this paper asymmetric algebraic riccati equations are analyzed in particular we derive a new parametrization of the set of solutions generalizing on the symmetric case the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related feedback matrices moreover the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo hamiltonian matrix associated with the equation we finally show that also the newly introduced parametrization is given by a homeomorphic map

explicit solutions of the discrete time lyapunov matrix equation and kalman yakubovich equations a general solution for the nonsquare nonsymmetric lyapunov matrix equation in a canonical form is presented the solution is shown to be a toeplitz matrix which may be calculated using the backwards levinson algorithm this solution is then applied to the kalman yakubovich equations to derive a method for generating strictly positive real functions via the positive real lemma this latter result has an application in system identification

centrosymmetric and centro skewsymmetric toeplitz plus hankel matrices and bezoutians abstract in the paper algebraic properties of centrosymmetric and centro skewsymmetric toeplitz plus hankel t h matrices are investigated it is shown that in the nonsingular case the inverse can be constructed with the help of the solutions of four related toeplitz systems with symmetric or skewsymmetric right hand sides in the singular case the related toeplitz matrices determine the kernel of the t h matrices finally centrosymmetric and centro skewsymmetric t h bezoutians are characterized

fast algorithms for close to toeplitz plus hankel systems and two sided linear prediction the low displacement rank definition of close to toeplitz ct matrices is extended to close to toeplitz plus hankel ctph matrices fast algorithms for solving ctph systems of equations are presented a matrix is defined as ctph if the sum of a ct matrix and a second ct matrix postmultiplied by an exchange matrix an equivalent definition in terms of uv rank is also given this definition is motivated by the application of the algorithms to two sided linear prediction tsp autocorrelation and covariance forms of tsp analogous to those for one sided linear prediction osp are defined the covariance form of tsp is solved using the ctph fast algorithms just as the covariance form of osp is solved using ct fast algorithms numerical examples show that tsp produces smaller residuals than osp and resolves sharp spectral peaks better than osp and that covariance tsp produces smaller residuals than autocorrelation tsp

an extremal problem for operators let a be a banach algebra f a compact set in the complex plane and h a function holomorphic in some neighborhood of the set f thus h a is meaningful for each element a e a whose spectrum a is contained in f and it is possible to evaluate the norm h a problem compute the supremum of the norms h a as a ranges over all elements of a with spectrum contained in f and whose norm does not exceed one that is compute sup h a a e a a f a 1 this problem was first formulated and treated by the author in the particular case where a is the algebra of all linear operators on a finite dimensional hilbert space and f is the disc z z r for a given positive number r 1 the paper discusses motivation connections with complex function theory convergence of iterative processes critical exponents and the infinite companion matrix

matrix decompositions using displacement rank and classes of commutative matrix algebras using the notion of displacement rank we look for a unifying approach to representations of a matrix a as sums of products of matrices belonging to commutative matrix algebras these representations are then considered in case a is the inverse of a toeplitz or a toeplitz plus hankel matrix some well known decomposition formulas for a gohberg semencul or kailath et al gader bini pan and gohberg olshevsky turn out to be special cases of the above representations new formulas for a in terms of algebras of symmetric matrices are studied and their computational aspects are discussed

analysis of the effect of impulse noise on multicarrier and single carrier qam systems the paper first shows the equivalence of the bernoulli gaussian impulse noise model in the discrete time domain to the continuous time model of poisson arriving delta functions with random area distributed according to the power rayleigh probability density function this equivalence is then used to develop a closed form expression for the probability of error for single carrier qam that is easily evaluated furthermore the performance of multicarrier modulation mcm is also analyzed using the same impulse noise model and it is shown that in most cases mcm performs better than single carrier systems specifically when the probability of an impulse is not too high and the impulse power is moderate

bezoutians and projections abstract with each polynomial p of degree n whose roots lie inside the unit disc we may associate the n dimensional space of all solutions of the recurrence relation whose coefficients are those of p considered as a subspace of 1 2 the main result consists in establishing a close relation between the bezoutian of two such polynomials of the same degree and the projection operator onto one of the corresponding spaces along the complement of the other the note forms a loose continuation of the author s investigations of the infinite companion matrix the generating function of the infinite companion matrix of a polynomial p appears thus as a particular case the corresponding bezoutian is that of the pair p and z n

on a new class of reproducing kernel spaces and a new generalization of the iohvidov laws abstract a new class of finite dimensional reproducing kernel spaces of m 1 vector valued analytic functions on a fairly general domain is introduced the reproducing kernels of these spaces have a special form which is based on an m m matrix valued function which is j unitary on the boundary of every invertible hermitian matrix can be interpreted as the gram matrix of a suitably chosen basis in such a space where m is equal to the appropriately defined displacement rank of the given matrix orthogonal decompositions of the space are developed in terms of schur complements of the matrix and factorizations of the much as in the classical cases wherein is either the unit disc or the half plane finally a new generalization of the iohvidov laws which extends earlier generalizations by the same authors is deduced as an application of the theory

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

on the bezoutian for polynomial matrices abstract the bezoutian b of two polynomial matrices can be described as a solution of a linear matrix equation this fact yields a new proof of the barnett factorization of b

hartley type algebras in displacement and optimization strategies abstract the hartley type ht algebras are used to face efficiently the solution of structured linear systems and to define low complexity methods for solving general nonstructured nonlinear problems displacement formulas for the inverse of a symmetric toeplitz matrix in terms of ht transforms are compared with the well known ammar gader formula the l qn unconstrained optimization methods which define hessian approximations by updating n n matrices from an algebra l can be implemented for l ht with an o n amount of memory allocations and o n log n arithmetic operations per step the l qn methods with the lowest experimental rate of convergence are shown to be linearly convergent

macroscopic behavior of microscopic oscillations in harmonic lattices via wigner husimi transforms we consider the dynamics of infinite harmonic lattices in the limit of the lattice distance tending to 0 we allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by fourier transform and linear algebra methods

detecting ofdm signals in alpha stable noise this paper analyzes various receiver schemes for orthogonal frequency division multiplexing ofdm transmission in impulsive noise we consider rayleigh block fading and model the noise process by additive white symmetric stable noise aws sn we begin by discussing maximum likelihood ml detection performance of baseband ofdm though optimal the computational cost increases exponentially with the number of carriers alternatively one may evaluate soft estimates of the transmitted symbol block and employ carrier wise detection to lower computational complexity we analyze such schemes under the frameworks of m estimation and compressed sensing theory moving on we highlight important rules that ensure the passband aws sn process is converted to a baseband form suitable for the discussed schemes finally it is shown that linear passband to baseband conversion actually reduces the signal to noise ratio snr at the receiver and that all these rules may be avoided by applying an estimation scheme directly on the passband samples

matrix representations of toeplitz plus hankel matrix inverses abstract inverses of toeplitz plus hankel matrices and more generally t h bezoutians are represented as sums of products of triangular toeplitz and hankel matrices the parameters occurring in these representations can be determined with the help of 1 solutions of fundamental equations 2 solutions of a certain homogeneous equation and 3 columns and rows of the inverse matrix

diagonal matrix solutions of a discrete time lyapunov inequality diagonal solutions of a lyapunov inequality for companion matrices are studied such solutions are required if states of a discrete time linear system are computed with a finite precision arithmetic

the group inverse of the transformation j x ax xb abstract the group inverse j of the sylvester transformation j x ax xb is provided that it exists represented in polynomial form j y ij c ij a i yb j in terms of the minimal polynomials of a and b in one of the representations the coefficient matrix c ij is expressed with the help of a new class of derived hankel matrices

more on loewner matrices abstract the study of loewner matrices is continued in a direction close to the recent work by fiedler ptak and vav in all results are stated for the general case of multiple interpolation nodes i e for generalized loewner matrices both the loewner bezoutian connection of antoulas and anderson and the loewner hankel connection of vav in in the general case which are re proved in a simple way provide a good opportunity for finding various properties of loewner matrices some of them being generalizations of results published by other authors for simple nodes only there are also results which are new especially the barnett like formulas and an intertwining relation characteristic for loewner matrices as well as the loewner matrix hankel vector connection in order to handle loewner matrices further on the authors introduce and deal with a class of so called r matrices it turns out that some known results concerning bezoutians hankel matrices and their products can in a natural way be lifted to the level of r matrices and of the products of loewner and r matrices

caratheodory sequence parametrizations of potapov normalized full rank jq elementary factors abstract full rank j q elementary factors play a key role in the context of the matricial version of the classical caratheodory interpolation problem extending work of i v kovalishina we derive various one to one correspondences between potapov normalized full rank j q elementary factors and finite nondegenerate q q caratheodory sequences

lyapunov equations for companion matrices abstract matrix equations of the form c a x xc t b y where ca and cb are companion matrices are considered in a first step 1 is reduced to an equation for which the solution is hankel and the right side has rank 2 at most in a second step a solvability condition and a general solution are given in terms of the polynomials a b and the solution of some polynomial congruence an application to resultant matrix equations is included

hankel matrices generated by the markov parameters of rational functions abstract finite hankel matrices s i j are considered for which s i are the markov parameters of a given rational function p a the problems of inversion and kernel description are reduced to the problem of solving some polynomial congruences in terms of p and a an application to hankel matrices of the form y t c i j a x where c a is a companion matrix is included

fast inversion algorithms of toeplitz plus hankel matrices the paper deals with the problems of fast inversion of matricesa t h wheret is toeplitz andh is hankel several algorithms are presented and compared among them algorithms working for arbitrary strongly nonsingular matricesa t h

large oscillations arising in a dispersive numerical scheme abstract we study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the hopf equation whenever the classical solution of the equation develops a singularity modulation equations are derived that describe period two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic however those equations have an elliptic region that may be entered by its solutions in a finite time after which the corresponding period two oscillations are seen to break down this kind of phenomenon has not been observed for integrable schemes the generation and propagation of period two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions modulation equations are also presented for period three oscillations numerical experiments are carried out that illustrate our analysis

modulation of multiphase waves in the presence of resonance the phenomenon of spatio temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem a simple family of exact solutions to the ablowitz ladik equations is found to be modulationally stable in some regimes this family of solutions is determined by fixing antiperiod 2 boundary conditions which determines two wavenumbers within the family of solutions the frequencies do not depend on amplitude this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation the family of solutions is described by four parameters two being actions that foliate the phase space and two being macroscopically observable functions of the phase constants the modulation of the actions is described by a closed hyperbolic system of first order equations which is consistent with the full set of four genus 1 modulation equations the modulation of the phase information easily observed due to the presence of two resonances is described by two more equations that are driven by the actions the results are confirmed by numerical experiments

generalized lyapunov equations matrices with displacement structure and generalized bezoutians abstract matrix equations of the form i 0 r j 0 s f ij a i xb j c are consider ed it is shown that solving such equations can be reduced to finding the solutions of certain well chosen auxiliary equations the latter equations are essentially simpler than the original equation and their solutions are matrices with a displacement structure or connected with generalized bezoutians in the case r s 1 the approaches given here along with some results on structured matrices and generalized bezoutians lead to explicit formulas for the solution

bezoutians and intertwining matrices abstract using the notion of an intertwining matrix introduced in a previous paper the authors show that the class of all n by n bezoutians compatible with a nonzero polynomial of degree n may be characterized by an intertwining relation as well as by four other conditions formulated in terms of the structure of these matrices

lyapunov bezout and hankel abstract given a polynomial f of degree n we denote by c its companion matrix and by s the truncated shift operator of order n we consider lyapunov type equations of the form x sxc w and x cxs w we derive some properties of these equations which make it possible to characterize bezoutian matrices as solutions of the first equation with suitable right hand sides w similarly for hankel and the second equation and to write down explicit expressions for these solutions this yields explicit factorization formulae for polynomials in c for the schur cohn matrix and for matrices satisfying certain intertwining relations as well as for bezoutian matrices

macroscopic lattice dynamics abstract fully nonlinear modulation equations are presented that govern the slow evolution of single phase harmonic wavetrains under a large family of spatially discrete nonintegrable flows among which is the discrete nonlinear schrodinger equation dnls these modulation equations are a pair of partial differential equations in conservation form that are hyperbolic for some data and elliptic for other data in some cases these equations are capable of dynamically changing type from hyperbolic to elliptic a phenomenon that has been associated with the modulational instability of the underlying wavetrain by putting the modulation equations in riemann invariant form one can select initial data that avoid this dynamic change of type numerical experiments demonstrating the theoretical results are presented

lyapunov equations and gram matrices if p is a polynomial with all roots inside the unit disc and c its companion matrix then the lyapunov equation x c xc p has a unique solution for every positive semidefinite matrix p we characterize sets of vectors x0 xn 1 and y0 yn 1 such that x g x0 xn 1 g y0 yn 1 1 geometrical connections between such bases and contractions with one dimensional defect spaces are established

on the use of cascade structure to correct impulsive noise in multicarrier systems we propose a new decoding algorithm based on a cascade structure for impulsive noise correction in multicarrier systems such as orthogonal frequency division multiplexing ofdm based systems this new cascade structure based algorithm has the advantage to lead on one hand to a reduced complexity implementation in an ofdm demodulator and on the other hand to overcome the lacks of other algorithms dedicated for impulsive noise correction to do so we start the derivation by investigating closed form analytical expressions of the a priori probabilities used in hypotheses tests that are necessary for the different steps needed for the execution of the cascade structure based decoding algorithm such closed form expressions for the a priori probabilities result in efficient setups of the threshold values used for the execution of the decoding algorithm furthermore we show that this cascade structure can be applied to efficiently cancel the impulsive noise in ofdm systems as the clipping event present in ofdm systems can be seen as an additive impulsive noise we prove that after introducing some minor modifications we can apply the proposed cascade structure to significantly reduce the peak to average power ratio papr level in such systems simulation results are given to illustrate the usefulness of the proposed algorithm results in practical ofdm based communication systems such as hiperlan 2

toward solution of matrix equation x af x b c abstract this paper studies the solvability existence of unique solution closed form solution and numerical solution of matrix equation x af x b c with f x x t f x x and f x x h where x is the unknown it is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard stein equations in the form of w a w b c where the dimensions of the coefficient matrices a b and c are the same as those of the original equation closed form solutions of equation x af x b c can then be obtained by utilizing standard results on the standard stein equation on the other hand some generalized stein iterations and accelerated stein iterations are proposed to obtain numerical solutions of equation x af x b c necessary and sufficient conditions are established to guarantee the convergence of the iterations

zero location by hermitian forms the singular case abstract the schur cohn criterion for the number of zeros of a polynomial inside and outside the unit disc fails if the polynomial has a pair of conjugate zeros or a zero on the unit circle the corresponding quadratic form is singular recently the authors have shown 5 that the classical schur cohn criterion may be deduced from a simple algebraic identity this yields not only a very simple proof but also a substantial generalization the method produces a whole family of quadratic forms which may be used for testing the zeros in the present paper the same algebraic identity is used to show that singularity of these quadratic forms is always due to the presence of pairs of conjugate zeros or zeros on the unit circle there is a method for ascertaining zero distribution in the singular case by differentiation we give a derivation of this test on the basis of our matrix theoritic treatment the second section deals with the same problem in the case of a general circle or half plane

the matrix equation axb gxd e over the quaternion field abstract using the method of complex representation this paper obtains necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix equation axb gxd e over the quaternion field extends w e roth s theorem and a jameson s theorem to the quaternion field and gives the representation of the solution

kernel structure of toeplitz plus hankel matrices abstract the structure of the kernel of block toeplitz plus hankel matrices r aj k bj k where aj and bj are the given p q blocks with entries from a given field is investigated it is shown that r corresponds to two systems of at most p q vector polynomials from which a basis of the kernel of r and all other toeplitz plus hankel matrices with the same parameters aj and bj can be built the main result is an analogue of a known kernel structure theorem for block toeplitz and block hankel matrices

on finding eigenvalue distribution of a matrix in several regions of the complex plane a direct method is proposed for determining eigenvalue distribution of a matrix with respect to several important regions of the complex plane these regions include half planes shifted half planes hyperbolas sectors quadrants imaginary axis region contained within two straight lines that pass through the orgin etc the method neither requires computation of the characteristic polynomial of the given matrix nor solution of any matrix equations the method seems to be more efficient than the eigenvalue and matrix equations methods

a fast algorithm for generalized hankel matrices arising in finite moment problems abstract consider an n n lower triangular matrix l whose i 1 st row is defined by the coefficients of the real polynomial p i x of degree i such that p i x is a set of orthogonal polynomials satisfying a standard three term recurrence relation if h is an n n real hankel matrix with nonsingular leading principal submatrices then h lhl t will be referred to as a strongly nonsingular hankel matrix with respect to the orthogonal polynomial basis p i x in this paper we develop an efficient o n 2 algorithm for the solution of a system of linear equations with a real symmetric coefficient matrix h which is a hankel matrix with respect to a suitable orthogonal polynomial basis this leads to an efficient method for computing polynomial approximations of an unknown function given its modified moments

on the jordan form of a family of linear mappings abstract let a and b be square matrices over a field f having their eigenvalues and in f and let g x y r sarsxrys be a polynomial over f assuming the jordan forms of a and b to be known the jordan form of r sarsar bs is determined when the partial derivatives g x and g y are nonzero at thus generalizing the classical cases g x y x y and g x y xy the particular cases g x y xk yl and g x y xkyl are also generalized by using properties of the partial hasse derivatives of g at the case where a and b are 2 2 or 3 3 matrices g being arbitrary is discussed exhaustively

on the inverses of toeplitz plus hankel matrices abstract it is well known that the inverses of hankel and toeplitz matrices can be represented as bezoutians of polynomials in the present note a bezoutian type formula for the inverses of toeplitz plus hankel matrices and a complete characterization of toeplitz plus hankel matrix inverses are given

dispersive regularization of the whitham equation for the toda lattice in this paper the averaged whitham equations for the slow modulations of multi phase wavetrains for the toda lattice are studied our main result is showing how the solution with step initial conditions may be regularized by choosing an appropriate riemann surface on which the whitham equation is defined

impulsive noise cancellation in multicarrier transmission a parallel between reed solomon codes in the complex field and multicarrier transmission using orthogonal frequency division multiplexing ofdm is first presented this shows that when the signal is sent over some channel composed of gaussian plus impulsive noise the impulsive noise can be removed by a procedure similar to channel decoding using information carried by the syndrome this result is first derived in a simple situation oversampled discrete multitone additive channel which is merely of theoretical interest in any case consecutive zeros in the output of the ofdm modulator do not correspond to real subcarriers pilot tones are transmitted for synchronization or channel estimation purposes these pilot tones are generally scattered among the information ones our approach is to use these pilot tones as syndromes in order to correct impulsive noise we show that the correction capacity is conditioned by the position of these pilot tones in the transmitted sequence a protection subsystem based on hypotheses tests is introduced after the decoding operation in order to detect malfunctions of this decoder the efficiency of this technique is corroborated with simulations in the slightly modified hiperlan2 context other extensions are then provided in order to increase the practical usefulness of the method

explicit expressions for bezoutians abstract the following commutator identity is proved u s v s v 1 s u 1 s here s is the n by n matrix of the truncated shift operator s i i 1 i 0 1 n 1 and u v are two polynomials of degree not exceeding n the reciprocal polynomial f 1 of a polynomial f of degree n is defined by f 1 z z n f 1 z the commutator identity is closely related to some properties of the bezoutian matrix of a pair of polynomials it is used to obtain the bezoutian matrix in the form of a simple expression in terms of s and s to demonstrate the advantage of this expression we show how it can be used to obtain simple proofs of some interesting corollaries

a schur algorithm and linearly connected processor array for toeplitz plus hankel matrices a levinson durbin type algorithm for solving toeplitz plus hankel t h linear systems of equations is used to induce a schur type algorithm for such systems a schur type algorithm is defined as one which efficiently computes the ldu decomposition of the matrix on the other hand levinson durbin type algorithms are defined as those algorithms which efficiently compute the udl decomposition of the inverse of a matrix it is shown that the schur algorithm so obtained is amenable to efficient implementation on a linearly connected array of processors in a manner which generalizes the results of s y kung and y h ku 1983 for symmetric toeplitz matrices specifically if t h is of order n then the schur algorithm runs on o n processors in o n time

an equation of lyapunov type abstract a new method of finding explicit solutions of lyapunov equations is described based on a lemma on one dimensional perturbations of invertible operators if y satisfies the equation y cyc bb for an appropriate vector b then x y 1 satisfies x c xc aa for a given vector a a concrete example with a 1 0 0 t is given

dft representations of toeplitz plus hankel bezoutians with application to fast matrixvector multiplication abstract representations for inverses of toeplitz plus hankel matrices and more general t h bezoutians with discrete fourier transformations are presented which can be used for fast matrix vector multiplication with the help of some of the formulas multiplication by a toeplitz plus hankel matrix inverse can be carried out with six dft s plus eight dft s for preprocessing which improves the so far best known result of seven dft s plus 10 dft s for preprocessing in e bozzo linear algebra appl 230 1995 127 150

bezoutian representation via vandermonde matrices abstract the result on benoutian representation via vandermonde matrices is provided with a proof which is based on polynomial models

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

on the history of the bezoutian and the resultant matrix abstract it is shown how the bezoutian and the resultant matrix evolved from euler s work in elimination theory

generalized companion matrices and matrix representations for generalized bezoutians abstract a generalization of companion matrices is presented as the starting point for developing a unified algebraic method to get matrix representations for generalized bezoutians special cases are discussed in particular well known formulas for hankel and toeplitz bezoutians are obtained also new matrix representations are given for example in a basis of orthogonal polynomials using corresponding christoffel darboux formulas

matrix representations of split bezoutians abstract inverses of symmetric or skewsymmetric toeplitz matrices as well as of centrosymmetric or centro skewsymmetric toeplitz plus hankel matrices can be represented as sums of two split bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors thus it is desirable to find matrix representations for split bezoutians b this is the main aim of the present paper recursion formulas for the entries of b are presented bases of very simple split bezoutians or of sparse matrices are constructed and b is represented as a corresponding linear combination moreover matrix representations of gohberg semencul type are established

inverse problems for positive hermitian block toeplitz matrices and nondegenerate schur sequences abstract using the concrete structure of the arov krein resolvent matrices connected with the matricial versions of the classical interpolation problems of schur and caratheodory several inverse problems are treated in this way former results of gohberg and lerer who studied the problem of determining a positive hermitian block toeplitz matrix given the first block column or the last block row of its inverse are extended in various directions moreover a new type of inverse problems for nondegenerate schur sequences is considered

algebraic methods for the solution of some linear matrix equations abstract an algebraic viewpoint permits the formulation of necessary and sufficient conditions for the existence of a unique solution to some linear matrix equations the theory developed is then used to express the solution in a finite series form and to prove a stability theorem

bernstein bezoutian matrices several computational and structural properties of bezoutian matrices expressed with respect to the bernstein polynomial basis are shown the exploitation of such properties allows the design of fast algorithms for the solution of bernstein bezoutian linear systems without never making use of potentially ill conditioned reductions to the monomial basis in particular we devise an algorithm for the computation of the greatest common divisor gcd of two polynomials in bernstein form a series of numerical tests are reported and discussed which indicate that bernstein bezoutian matrices are much less sensitive to perturbations of the coefficients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the bernstein and the power basis

matrix representations of bezoutians abstract an approach is presented to get matrix representations for classical and more general r bezoutians with entries from a noncommutative ring formulas due to ammar and gader barnett gohberg and semencul gohberg and heinig kravitsky and russakovskii lerer and tismentsky ptak and wimmer as well as some generalizations and modifications are obtained in a unified way

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

a necessary condition on the location of pilot tones for maximizing the correction capacity in ofdm systems this paper presents a new view of the bose chaudhuri hocquengem bch code through the addition of some flexibility to the syndromes distribution in the transmitted sequence in order to get this flexibility we derive a necessary condition nc on the syndromes distribution for decoding bch codes which includes the already known hartmann tzeng proposition this nc is essentially deduced from the decoding process of bch code and is related to the locator polynomial and the requested constraints to guarantee a maximal error correction capacity the obtained results have the advantage to be applicable for any considered field finite or not furthermore we prove that when the correction capacity is equal to 2 or 3 the obtained nc becomes also sufficient this result is very useful in some practical transmission systems such as orthogonal frequency division multiplexing systems once the pilot tones considered in such systems verify the necessary and sufficient condition it becomes possible to both reduce the peak to average power rate and correct the impulse noise present in such multicarrier systems the usefulness of the presented analysis and the exploitation of the derived condition on the pilot tones distribution is illustrated by simulation results in the case of the hiperlan2 system

the discrete lyapunov equation in controllable canonical form a new method of treating lyapunov equations is proposed if d and d are two self adjoint one dimensional matrices related in a certain way then x c ast xc d if and only if x 1 cx 1 c ast d as an application a generalization of a recent result is given if f is the vector f 0 0 0 1 t then the solution of x cxc ast ff ast is shown to be inverse of the schur cohn matrix

the matrix equation xa bx r and its applications abstract we study the well known sylvester equation xa bx r in the case when a and b are given and r is known up to its first n 1 rows we prove new results on the existence and uniqueness of x our results essentially state that in case a is a nonderogatory matrix there always exists a solution to this equation a solution is uniquely determined by its first row x 1 and there is an interesting relationship between x 1 and the rows of r we also give a complete characterization of the nonsingularity of x in this case as applications of our results we develop direct methods for constructing symmetrizers and commuting matrices computing the characteristic polynomial of a matrix and finding the numbers of common eigenvalues between a and b some well known important results on symmetrizers bezoutians and inertia are recovered as special cases

the bezoutian the hankel matrix output feedback and system eigenstructure abstract given a minimal linear time invariant system for which the transfer matrix has a right rp 1 and a left d 1 n coprime matrix fraction description an inverse relation is established between the block hankel matrix formed from the markov parameters of the eigenstructure of p and d and the generalized bezoutian matrix formed from the polynomial matrices r p n and d furthermore it is shown that all eigenstructures that are assignable by output feedback form the same markov parameters and this suggests a new approach to eigenstructure assignment

fast triangular factorization of the sum of quasi toeplitz and quasi hankel matrices abstract the literature contains several recent fast algorithms for the triangular factorization of strongly regular toeplitz plus hankel matrices in this paper we study the rather more general sum of quasi toeplitz and quasi hankel matrices both hermitian and non hermitian quasi toeplitz and quasi hankel matrices are those that are congruent to toeplitz and hankel matrices in a special sense the derivation is based on the concept of displacement structure and its intimate relation to the schur reduction procedure for triangular factorization various special cases covering displacement ranks from two to eight are considered several other problems e g factorization of the inverse matrix solution of exact or overdetermined linear systems can be reduced to the direct factorization problem

block sparse impulsive noise reduction in ofdm systems a novel iterative bayesian approach using a novel block iterative bayesian algorithm block iba this paper presents a new impulsive noise reduction method for ofdm systems the method utilizes the guard band null subcarriers and data subcarriers for the impulsive noise estimation and cancellation unlike some other general ofdm transceivers which use time domain interleaving tdi to cancel impulsive noise we design a specific receiver for bursty impulsive noise channels that removes the delay due to tdi and saves memory space the block iba first estimates the variance and the transition matrix of markov chain model for the impulsive noise it then iteratively estimates the amplitudes and positions of the block sparse impulsive noise using the steepest ascent based expectation maximization em and optimally selects the nonzero elements of the block sparse impulsive noise by adaptive thresholding numerical experiments show that the proposed receiver outperforms existing receivers under the block sparse impulsive noise environment

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

linear matrix equations the module theoretic approach abstract the matrix equation fikaixbk c is discussed based on the algebraic approach of djaferis and mitter