chebyshev hankel matrices and the splitting approach for centrosymmetric toeplitz plus hankel matrices abstract centrosymmetric toeplitz plus hankel matrices are investigated on the basis of their splitting property which is their similarity to the direct sum of two special toeplitz plus hankel matrices these matrices can be considered as hankel matrices moment matrices in bases of chebyshev polynomials and are called chebyshev hankel matrices chebyshev hankel matrices have similar properties like hankel matrices this concerns inversion formulas and fast algorithms a superfast algorithm for solving chebyshev hankel and centrosymmetric toeplitz plus hankel systems is presented that is based on real trigonometric transforms the main tool of investigation is the interpretation of chebyshev hankel matrices as matrices of restricted multiplication operators with respect to chebyshev bases

infinitely divisible nonnegative matrices m matrices and the embedding problem for finite state stationary markov chains abstract this paper explicitly details the relation between m matrices nonnegative roots of nonnegative matrices and the embedding problem for finite state stationary markov chains the set of nonsingular nonnegative matrices with arbitrary nonnegative roots is shown to be the closure of the set of matrices with matrix roots in im the methods presented here employ nothing beyond basic matrix analysis however it answers a question regarding m matrices posed over 30 years ago and as an application a new characterization of the set of all embeddable stochastic matrices is obtained as a corollary

on copositive matrices abstract a criterion for copositive matrices is given and for n 3 the set of all copositive matrices is determined in terms of matrix elements copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations

graph theoretic methods for matrix completion problems abstract a pattern is a list of positions in an n n real matrix a matrix completion problem for the class of matrices asks whether every partial matrix whose specified entries are exactly the positions of the pattern can be completed to a matrix we survey the current state of research on matrix completion problems for many subclasses of p 0 matrices including positive definite matrices m matrices inverse m matrices p matrices and matrices defined by various sign symmetry and positivity conditions on p 0 and p matrices graph theoretic techniques used to study completion problems are discussed several new results are also presented including the solution to the m 0 matrix completion problem and the sign symmetric p 0 matrix completion problem

some properties of schur complements and diagonal schur complements of diagonally dominant matrices in this paper we prove that the diagonal schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix the same holds for the diagonal schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular h matrix we point out that under certain assumptions the diagonal schur complement of a strictly doubly doubly product diagonally dominant matrix is also strictly doubly doubly product diagonally dominant further we provide the distribution of the real parts of eigenvalues of a diagonal schur complement of h matrix we also show that the schur complement of a diagonally dominant matrix is not always diagonally dominant by a numerical example and then obtain a sufficient condition to ensure that the schur complement of a diagonally dominant matrix is diagonally dominant

looking for nonnegative solutions of a leontief dynamic model let a be an n n nonnegative real matrix we know that a is monotone if it has a nonnegative inverse moreover if a has an appropriate generalized inverse then we can obtain nonnegative solutions for the linear system ax b where b 0 in this paper using properties about nonnegatives matrices we present sufficient conditions in order that the leontief price model i f p k 1 1 r k fp k gp0 k 1 a0 where f i at 1bt and g i at 1 have nonnegative solutions

the algebraic approach to the discrete cosine and sine transforms and their fast algorithms it is known that the discrete fourier transform dft used in digital signal processing can be characterized in the framework of the representation theory of algebras namely as the decomposition matrix for the regular module mathbb c z n mathbb c x x n 1 this characterization provides deep insight into the dft and can be used to derive and understand the structure of its fast algorithms in this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of chebyshev polynomials then we derive most of their known algorithms by pure algebraic means we identify the mathematical principle behind each algorithm and give insight into its structure our results show that the connection between algebra and digital signal processing is stronger than previously understood

inverse no matrices abstract results similar to those of thomas l markham concerning inverse m matrices are shown to hold for inverse n o matrices a more general result and an alternate method of proof using schur complements are given for one of markham s theorems conditions are given to transform a nonpositive matrix to an n o amtrix by multiplication by householder transformations also an lu factorization of an inverse n o matrix is given

renyi divergence and kullback leibler divergence renyi divergence is related to renyi entropy much like kullback leibler divergence is related to shannon s entropy and comes up in many settings it was introduced by renyi as a measure of information that satisfies almost the same axioms as kullback leibler divergence and depends on a parameter that is called its order in particular the renyi divergence of order 1 equals the kullback leibler divergence we review and extend the most important properties of renyi divergence and kullback leibler divergence including convexity continuity limits of sigma algebras and the relation of the special order 0 to the gaussian dichotomy and contiguity we also show how to generalize the pythagorean inequality to orders different from 1 and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs for all orders and present several other minimax results

the schur complements of generalized doubly diagonally dominant matrices as is known the schur complements of diagonally dominant matrices are diagonally dominant the same is true of doubly diagonally dominant matrices the purpose of this paper is to extend the results to the generalized doubly diagonally dominant matrices a proper subset of h matrices that is we show that the schur complement of a generalized doubly diagonally dominant matrix is a generalized doubly diagonally dominant matrix

the quasi principal rank characteristic sequence abstract a minor of a matrix is quasi principal if it is a principal or an almost principal minor the quasi principal rank characteristic sequence qpr sequence of an n n symmetric matrix is introduced which is defined as q 1 q 2 q n where q k is a s or n according as all some but not all or none of its quasi principal minors of order k are nonzero this sequence extends the principal rank characteristic sequences in the literature which only depend on the principal minors of the matrix a necessary condition for the attainability of a qpr sequence is established using probabilistic techniques a complete characterization of the qpr sequences that are attainable by symmetric matrices over fields of characteristic 0 is given

on schur complement of block diagonally dominant matrices abstract it is well known that the schur complements of strictly diagonally dominant matrices are strictly diagonally dominant d carlson t markham schur complements of diagonally dominant matrices czech math j 29 104 1979 246 251 the same is true of generalized strictly diagonally dominant matrices jianzhou liu yungqing huang some properties on schur complements of h matrix and diagonally dominant matrices linear algebra appl 389 2004 365 380 in this paper this result is extended to the block strictly diagonally dominant matrices and the generalized block strictly diagonally dominant matrices that is it is shown that the schur complement of a block strictly diagonally dominant matrix is a block strictly diagonally dominant matrix and so is the schur complement of a generalized block strictly diagonally dominant matrix

sign patterns of inverse nonnegative matrices abstract we determine here the 0 sign patterns which occur among the inverses of nonsingular entrywise nonnegative matrices these results complete a sequence of work which began with the characterization of all possible sign patterns which occur among inverses of positive matrices and the characterization in this general case is more involved to state than the earlier ones

on mean convergence of extended lagrange interpolation abstract lagrange interpolation to any continuous function on 1 1 at the zeros of orthogonal polynomials is known to converge in the mean here following bellen we study mean convergence of lagrange interpolation on an extended set of nodes that includes in addition to the n zeros of the orthagonal relative to some positive weight function w polynomial n of degree n other n 1 nodes which in turn are zeros of an orthogonal polynomial gp n 1 of degree n 1 corresponding to the weight function w n 2 n w a sufficient criterion of bellen for mean convergence as n of such extended lagrange interpolation for arbitrary continuous functions is shown to fail for chebyshev weight functions of the first third and fourth kind it holds trivially for chebyshev weights of the second kind based on extensive computations it is conjectured on the other hand that the criterion is satisfied for certain jacobi weights with parameters and suitably restricted necessary conditions for mean convergence due to erd s and turan are shown to be violated for the three kinds of chebyshev weights mentioned above for smooth functions a comparison is made of the speed of convergence of simple vs extended lagrange interpolation

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

closure properties of certain positivity classes of matrices under various algebraic operations abstract we present a table indicating whether or not each of five positivity classes of matrices positive definite hermitian matrices m matrices inverse m matrices totally positive matrices and inverse totally positive matrices is closed under each of seven algebraic operations conventional multiplication addition powers extraction of roots hadamard multiplication the hadamard product of one element and the inverse of another and lu factorization three of these 35 facts seem not to have been previously known and key examples and references are given

inequalities for m matrices and inverse m matrices in this paper we establish some determinantal inequalities concerning m matrices and inverse m matrices the main results are as follows 1 if a aij is either an n n m matrix or inverse m matrix then for any permutation i1 i2 in of 1 2 n a deta i 1naii s 2n1 ai1i2 ais 1isaisi1 ai1i1ai2i2 aisis b deta i 1naii if and only if a is essentially triangular 2 if a aij is an n n m matrix b bij is an n n inverse m matrix a b denotes the hadamard product of a and b then a b is an m matrix and for any permutation i1 i2 in of 1 2 n det a b det ab s 2naisisdeta i1 i2 is 1 deta i1 is 1 is bisisdetb i1 i2 is 1 detb i1 is 1 is 1

on classes of inverse z matrices abstract recently a classification of matrices of class z was introduced by fiedler and markham this classification contains the classes of m matrices and the classes of n 0 and f 0 matrices studied by fan g johnson and smith the problem of determining which nonsingular matrices have inverses which are z matrices is called the inverse z matrix problem for special classes of z matrices such as the m and n 0 matrices there exist at least partial results i e special classes of matrices have been introduced for which the inverse of such a matrix is an m matrix or an n 0 matrix here we define a system of classes of matrices for which the inverse of each matrix of each class belongs to one class of the classification of z matrices defined by fiedler and markham moreover certain properties of the matrices of each class are established e g inequalities for the sum of the entries of the inverse and the structure of certain schur complements we also give a necessary and sufficient condition for regularity the class of inverse n 0 matrices given here generalizes the class of inverse n 0 matrices discussed by johnson all results established here can be applied to a class of distance matrices which corresponds to a nonarchimedean metric this metric arises in p adic number theory and in taxonomy

the dominant degree and disc theorem for the schur complement of matrix the theory of schur complement plays an important role in many fields such as control theory and computational mathematics in this paper we obtain some estimates for the diagonally c diagonally and product c diagonally dominant degree on the schur complement of matrices which improve some relative results as application we present some bounds for the eigenvalues of schur complement by the entries of the original matrix instead of those of the schur complement particularly we obtain that the eigenvalues of the schur complements are located in the gerschgorin circles of the original matrices under certain conditions

on divergences and informations in statistics and information theory the paper deals with the f divergences of csiszar generalizing the discrimination information of kullback the total variation distance the hellinger divergence and the pearson divergence all basic properties of f divergences including relations to the decision errors are proved in a new manner replacing the classical jensen inequality by a new generalized taylor expansion of convex functions some new properties are proved too e g relations to the statistical sufficiency and deficiency the generalized taylor expansion also shows very easily that all f divergences are average statistical informations differences between prior and posterior bayes errors mutually differing only in the weights imposed on various prior distributions the statistical information introduced by de groot and the classical information of shannon are shown to be extremal cases corresponding to alpha 0 and alpha 1 in the class of the so called arimoto alpha informations introduced in this paper for 0 alpha 1 by means of the arimoto alpha entropies some new examples of f divergences are introduced as well namely the shannon divergences and the arimoto alpha divergences leading for alphauarr1 to the shannon divergences square roots of all these divergences are shown to be metrics satisfying the triangle inequality the last section introduces statistical tests and estimators based on the minimal f divergence with the empirical distribution achieved in the families of hypothetic distributions for the kullback divergence this leads to the classical likelihood ratio test and estimator

information divergence and risk for binary experiments we unify f divergences bregman divergences surrogate regret bounds proper scoring rules cost curves roc curves and statistical information we do this by systematically studying integral and variational representations of these objects and in so doing identify their representation primitives which all are related to cost sensitive binary classification as well as developing relationships between generative and discriminative views of learning the new machinery leads to tight and more general surrogate regret bounds and generalised pinsker inequalities relating f divergences to variational divergence the new viewpoint also illuminates existing algorithms it provides a new derivation of support vector machines in terms of divergences and relates maximum mean discrepancy to fisher linear discriminants

inverse m matrix completions of patterns omitting some diagonal positions abstract let n 1 n which subsets q of n n have the property that whenever a ij for i j q form a partial inverse m matrix i e a ij 0 and if l l q then the inverse of a ij i j l is an m matrix a ij can be defined for i j q so that the inverse of a a ij is an m matrix in terms of a digraph g with vertex set n and arcs i j q with i j our results say 1 it is necessary that the induced subdigraph of any alternate path to an arc be a clique 2 a cycle need not be contained in a clique if i i q for at least one vertex i of 3 when n 4 it is necessary and sufficient that condition 1 is true and every simple cycle has a vertex i with i i q and 4 the general question can be reduced to the case in which g is strongly connected much as it has been when q contains i i for every i n

nonnegative matrices with power invariant zero patterns abstract if a is a nonsingular m matrix the elements of the sequence a k all have the same zero pattern using the drazin inverse we show that a similar zero pattern invariance property holds for a class of matrices which is larger than the generalized m matrices

on perron complements of inverse n0 matrices abstract in this paper we show that the perron complements of irreducible n 0 matrices are n 0 matrices we also demonstrate the perron complements of irreducible inverse n 0 matrices are inverse n 0 matrices with a little restriction in addition we give some related inequalities about inverse n 0 matrices

on the inverse of a special schur complement abstract the inverse of a schur complement is a very useful tool in many algorithms for the computation of the matrix inversion in this paper we study the inverse of a special schur complement cd 1 b we proved that always exist some x and y such that xdy is a inverse of cd 1 b furthermore using minimal rank properties we give some explicit expressions for x y and the inverse of cd 1 b numerical examples also given

principal pivot transforms properties and applications the principal pivot transform ppt is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right hand side of the system the notion of the ppt is encountered in mathematical programming statistics and numerical analysis among other areas the purpose of this paper is to draw attention to the main properties and uses of ppts make some new observations and motivate further applications of ppts in matrix theory special consideration is given to ppts of matrices whose principal minors are positive 2000 elsevier science inc all rights reserved

on the inverse mean first passage matrix problem and the inverse m matrix problem abstract the inverse mean first passage time problem is given a positive matrix m r n n then when does there exist an n state discrete time homogeneous ergodic markov chain c whose mean first passage matrix is m the inverse m matrix problem is given a nonnegative matrix a then when is a an inverse of an m matrix the main thrust of this paper is to show that the existence of a solution to one of the problems can be characterized by the existence of a solution to the other in so doing we extend earlier results of tetali and fiedler

on the renyi divergence joint range of relative entropies and a channel coding theorem this paper starts by considering the minimization of the renyi divergence subject to a constraint on the total variation distance based on the solution of this optimization problem the exact locus of the points bigl d q p 1 d q p 2 bigr is determined when p 1 p 2 and q are arbitrary probability measures which are mutually absolutely continuous and the total variation distance between p 1 and p 2 is not below a given value it is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet this characterization yields a geometric interpretation of the minimal chernoff information subject to a constraint on the variational distance this paper also derives an exponential upper bound on the performance of binary linear block codes or code ensembles under maximum likelihood decoding its derivation relies on the gallager bounding technique and it reproduces the shulman feder bound as a special case the bound is expressed in terms of the renyi divergence from the normalized distance spectrum of the code or the average distance spectrum of the ensemble to the binomially distributed distance spectrum of the capacity achieving ensemble of random block codes this exponential bound provides a quantitative measure of the degradation in performance of binary linear block codes or code ensembles as a function of the deviation of their distance spectra from the binomial distribution an efficient use of this bound is considered

generalized ultrametric matrices a class of inverse m matrices abstract recently marti nez michon and san marti n introduced the new class of symmetric strictly ultrametric matrices they proved that the inverse of a strictly ultrametric matrix is a strictly row and strictly column diagonally dominant stieltjes matrix here we generalize their result by introducing a class of nonsymmetric matrices calledgeneralized ultrametric matrices we give a necessary and sufficient condition for the regularity of these matrices and prove that the inverse of a nonsingular generalized ultrametric matrix is a row and column diagonally dominant m matrix we establish that a nonnegative matrix is a generalized ultrametric matrix if and only if the matrix is a certain sum of at most rank two matrices moreover we give a characterization of generalized ultrametric matrices based on weighted trees the entries of generalized ultrametric matrices then arise as certain distances between the leaves and the root of the tree

near minimax complex approximation by four kinds of chebyshev polynomial expansion abstract functions of the form w z f z with f analytic and w z 1 z 2 1 1 2 z 1 1 2 or z 1 1 2 are approximated in the ellipse r z z 2 1 1 2 r by w z pn z where pn is a polynomial of degree n here pn is obtained by the expansion of f in chebyshev polynomials of the first second third or fourth kinds corresponding to the above four respective weight functions bounds are established and computed for the norms on r of the corresponding projections thus confirming that all resulting approximations are near minimax within relative distances asymptotically proportional to 4 2 ln n and extending a known result geddes 1978 for w z 1 and first kind chebyshev polynomials

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

characterizations of inverse m matrices with special zero patterns in this paper we provide some characterizations of inverse m matrices with special zero patterns in particular we give necessary and sufficient conditions for k diagonal matrices and symmetric k diagonal matrices to be inverse m matrices in addition results for triadic matrices tridiagonal matrices and symmetric 5 diagonal matrices are presented as corollaries

notes on completely positive matrices abstract let a be a n n symmetric matrix and in the closure of inverse m matrices then a can be factored as a bb t for some nonnegative lower triangular n n matrix b and cp rank a n if a is a positive semidefinite 0 1 matrix then a is completely positive and cp rank a rank a if a is a nonnegative symmetric h matrix then a is completely positive and cp rank a n n 1 2 n n where is the number of connected components of the graph g a

the schur complement and the inverse m matrix problem abstract it is well known that if m is a nonnegative nonsingular inverse m matrix and if a is a nonsingular block in the upper left hand corner then the schur complement of a in m m a is an inverse m matrix the converse of this is generally false in this paper we give added restrictions on m or a to insure the converse and give some necessary and sufficient conditions for hmh 1 where h i i to be an m matrix

the linear order complementarity problem the classical complementarity problem in euclidean space can be viewed alternatively as a variational inequality or as a lattice orthogonality problem generalizations of the former have been extensively studied but infinite dimensional analogues of the latter have been largely ignored moreover as we show many well known results about the classical complementarity problem are more appropriately viewed order theoretically this is particularly true of least element solutions which are central to the present study of order complementarity in vector lattices we emphasize that the lattice theoretic descriptions we employ are very useful even in the standard finite dimensional setting

tight bounds for symmetric divergence measures and a refined bound for lossless source coding tight bounds for several symmetric divergence measures are derived in terms of the total variation distance it is shown that each of these bounds is attained by a pair of two or three element probability distributions an application of these bounds for lossless source coding is provided refining and improving a certain bound by csiszar another application of these bounds has been recently introduced by yardi et al for channel code detection

approximating cauchy type singular integral by an automatic quadrature scheme an automatic quadrature scheme is developed for the approximate evaluation of the product type indefinite integral q f x y c x y r t k t c f t dt 1 x y 1 1

tridiagonal and upper triangular inverse m matrices abstract in this paper we characterize the nonnegative nonsingular tridiagonal matrices belonging to the class of inverse m matrices we give a geometric equivalence for a nonnegative nonsingular upper triangular matrix to be in this class this equivalence is extended to include some reducible matrices

new algorithms for solving high even order differential equations using third and fourth chebyshev galerkin methods this paper is concerned with spectral galerkin algorithms for solving high even order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions the proposed algorithms are extended to solve two dimensional high even order differential equations the key to the efficiency of these algorithms is to construct compact combinations of chebyshev polynomials of the third and fourth kinds as basis functions the algorithms lead to linear systems with specially structured matrices that can be efficiently inverted numerical examples are included to demonstrate the validity and applicability of the proposed algorithms and some comparisons with some other methods are made

inverse h matrices abstract the notion of inverse m matrices has been quite well studied and many of their properties have been documented in the literature in this article a proposal to consider the larger class of inverse h matrices is made first certain special types of real matrices are identified as inverse h matrices while these matrices are not inverse m matrices in general their definitions are inspired by matrices that are known to be inverse m matrices in the rest of the article motivated by results that hold for inverse m matrices the endeavor will be to attempt at proving corresponding results for inverse h matrices it is shown that for the full class of inverse h matrices many of these extensions are false nevertheless two subclasses of real inverse h matrices are identified where these generalizations are shown to hold

the degree of an exact order matrix the classes of exact order k matrices for any positive integer k were defined and studied by mohan parthasarathy and sridhar mohan s r t parthasarathy r sridhar 1994 the linear complementarity problem with exact order matrices math oper res 19 618 644 here we prove results on the linear complementarity problem lcpq m for m belonging to the class of exact order k k 3 using the concepts of degree theory our main result in this paper consists in proving that a matrix m rn n of exact order k for any positive integer n k 3 belongs to the class q if and only if the degree of m is either 1 or 1 also a complete characterization of exact order 2 matrices is presented in terms of their inverse structure

on diagonal schur complements of block diagonally dominant matrices it is known that the diagonal schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices j z liu y q huang some properties on schur complements of h matrices and diagonally dominant matrices linear algebra appl 389 2004 365 380 and the same is true for nonsingular h matrices j z liu j c li z t huang x kong some properties of schur complements and diagonal schur complements of diagonally dominant matrices linear algebra appl 428 2008 1009 1030 in this paper we research the properties on diagonal schur complements of block diagonally dominant matrices and prove that the diagonal schur complements of block strictly diagonally dominant matrices are block strictly diagonally dominant matrices and the same holds for generalized block strictly diagonally dominant matrices

monotonicity and positivity of coefficients of power series expansions associated with newton and halley methods for the matrix pth root abstract if a is a matrix with all eigenvalues in the disk z 1 1 the principal pth root of a can be computed by newton s method or halley s method the study of newton s method and halley s method for the matrix pth root can be done through a study of power series expansions of some sequences of scalar functions in this paper we prove monotonicity results for the coefficients in the power series expansions for both newton s method and halley s method and for all integer p 2 the sign patterns of these coefficients can be seen directly from the monotonicity results we then use these monotonicity results and their consequences to obtain some new error estimates in the matrix case to obtain a monotonic convergence result when a is a nonsingular m matrix and to obtain a structure preserving result when a is a nonsingular m matrix or a real nonsingular h matrix with positive diagonal entries

matrices and the linear complementarity problem the paper is a collection of results on the linear complementarity problem q m the results are stated in terms of the matrix m

matrices with a transitive graph and inverse m matrices abstract the relationship between inverse m matrices and matrices whose graph is transitive is studied the results are applied to obtain a new proof of the characterization due to m lewin and m neumann of 0 1 inverse m matrices

criteria for sign regularity of sets of matrices abstract we consider matrix intervals with respect to a certain checkerboard partial ordering we show that every real matrix contained in a matrix interval is sign regular if two special matrices taken from that matrix interval are sign regular

inverse m matrix inequalities and generalized ultrametric matrices abstract we use weighted directed graphs to introduce a class of nonnegative matrices which under a simple condition are inverse m matrices we call our class the generalized ultrametic matrices since it contains the class of symmetric ultrametric matrices and some unsymmetric matrices we show that a generalized ultrametric matrix is the inverse of a row and column diagonally dominant m matrix if and only if it contains no zero row and no two of its rows are identical this theorem generalizes the known result that a symmetric strictly ultrametric matrix is the inverse of a strictly diagonally dominant m matrix we also present inequalities and conditions for equality among the entries of the inverse of a row diagonally dominant m matrix some of these inequalities and conditions for equality generalize results of stieltjes on inverses of symmetric diagonally dominant m matrices

on the hadamard product of inverse m matrices abstract we investigate the hadamard product of inverse m matrices and present two classes of inverse m matrices that are closed under the hadamard multiplication in the end we give some inequalities on the fan product of m matrices and schur complements

sharp inequalities for f divergences f divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability mathematical statistics and information theory such as kullback leibler divergence chi squared divergence squared hellinger distance total variation distance and so on in this paper we study the problem of maximizing or minimizing an f divergence between two probability measures subject to a finite number of constraints on other f divergences we show that these infinite dimensional optimization problems can all be reduced to optimization problems over small finite dimensional spaces which are tractable our results lead to a comprehensive and unified treatment of the problem of obtaining sharp inequalities between f divergences we demonstrate that many of the existing results on inequalities between f divergences can be obtained as special cases of our results we also improve on some existing non sharp inequalities

the symmetric m matrix and symmetric inverse m matrix completion problems abstract the symmetric m matrix and symmetric m 0 matrix completion problems are solved and results of johnson and smith linear algebra appl 290 1999 193 are extended to solve the symmetric inverse m matrix completion problem 1 a pattern i e a list of positions in an n n matrix has symmetric m completion i e every partial symmetric m matrix specifying the pattern can be completed to a symmetric m matrix if and only if the principal subpattern r determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete or in graph theoretic terms if and only if each component of the graph of r is a clique 2 a pattern has symmetric m 0 completion if and only if the pattern is permutation similar to a pattern that is block diagonal with each diagonal block either complete or omitting all diagonal positions or in graph theoretic terms if and only if every principal subpattern corresponding to a component of the graph of the pattern either omits all diagonal positions or includes all positions 3 a pattern has symmetric inverse m completion if and only if its graph is block clique and no diagonal position is omitted that corresponds to a vertex in a graph block of order 2 the techniques used are also applied to matrix completion problems for other classes of symmetric matrices

on inverse m matrices abstract we generalize a previous result of lewin and neumann and present a characterization of inverse m matrices the approach is mainly graph theoretical

chebyshev polynomials of the second third and fourth kinds in approximation indefinite integration and integral transforms abstract chebyshev polynomials of the third and fourth kinds orthogonal with respect to 1 x 1 2 1 x 1 2 and 1 x 1 2 1 x 1 2 respectively on 1 1 are less well known than traditional first and second kind polynomials we therefore summarise basic properties of all four polynomials and then show how some well known properties of first kind polynomials extend to cover second third and fourth kind polynomials specifically we summarise a recent set of first second third and fourth kind results for near minimax constrained approximation by series and interpolation criteria then we give new uniform convergence results for the indefinite integration of functions weighted by 1 x 1 2 or 1 x 1 2 using third or fourth kind polynomial expansions and finally we establish a set of logarithmically singular integral transforms for which weighted first second third and fourth kind polynomials are eigenfunctions

the schur complements of γ diagonally and product γ diagonally dominant matrix and their disc separation in this paper we obtain some estimates for the diagonally and product diagonally dominant degree of the schur complement of matrices as application we present some bounds for the eigenvalues of schur complement by the entries of the original matrix based on these results we give a kind of iteration called the schur based iteration which can solve large scale linear systems though reducing the order by the schur complement and can compute out the results faster

wave propagation in bifurcated waveguides of square lattice strips the propagation of waves in square lattice strips as waveguides with discrete analogues of hard and soft outer boundaries is considered in the presence of a bifurcating lattice row which is either soft or hard by an application of the discrete fourier transform the related boundary value problem for both types of the bifurcated waveguides is formulated as a discrete wiener hopf equation the exact solution of the latter is found in the form of a contour integral on the unit circle in complex plane closed form expressions for the approximate solution in the far field are provided using the normal mode expansion a general expression for the reflectance and transmittance is obtained the sum in the absence of dissipation is established as unity it is shown that the limiting wave field as the waveguide width tends to infinity coincides with that on an infinite lattice the proof is presented in the dissipative case the discrete paradigm of the bifurcated waveguides introduced in the paper has natural applications in engineering and science

z matrices and inverse z matrices abstract we consider z matrices and inverse z matrices i e those nonsingular matrices whose inverses are z matrices recently fiedler and markham introduced a classification of z matrices this classification directly leads to a classification of inverse z matrices among all classes of z matrices and inverse z matrices the classes of m matrices n 0 matrices f 0 matrices and inverse m matrices inverse n 0 matrices and inverse f 0 matrices respectively have been studied in detail here we discuss each single class of z matrices and inverse z matrices as well as considering the whole classes of z matrices and inverse z matrices we establish some common properties of the classes such as eigenvalue bounds and determinant inequalities and we give a new characterization of some classes of z matrices and inverse z matrices moreover we prove that other classes besides those of m matrices n 0 matrices and f 0 matrices consist of matrices whose determinants have the same sign some of the results generalize known results for m matrices n 0 matrices and f 0 matrices and for inverse m matrices inverse n 0 matrices and inverse f 0 matrices however we also show that some properties of the specific classes mentioned above do not hold for all classes of z matrices and inverse z matrices

geometric aspects of the symmetric inverse m matrix problem abstract we investigate the symmetric inverse m matrix problem from a geometric perspective the central question in this geometric context is which conditions on the k dimensional facets of an n simplex s guarantee that s has no obtuse dihedral angles the simplest of such conditions is that if all triangular facets of s are equilateral then s is regular and thus nonobtuse first we study the properties of an n simplex s whose k facets are all nonobtuse and generalize some classical results by fiedler 14 we prove that if all n 1 facets of an n simplex s are nonobtuse each makes at most one obtuse dihedral angle with another facet this helps to identify a special type of tetrahedron which we will call sub orthocentric with the property that if all tetrahedral facets of s are sub orthocentric then s is nonobtuse rephrased in the language of linear algebra this constitutes a purely geometric proof of the fact that each symmetric ultrametric matrix 28 is the inverse of a weakly diagonally dominant m matrix the geometric proof provides valuable insights that supplement the discrete measure theoretic 24 and the linear algebraic 25 proof the review papers 20 21 support our believe that the linear algebraic perspective on the inverse m matrix problem dominates the literature the geometric perspective however connects sign properties of entries of inverses of a symmetric positive definite matrix to the dihedral angle properties of an underlying simplex and enables an explicit visualization of how these angles and signs can be manipulated this will serve to formulate purely geometric conditions on the k facets of an n simplex s that may render s nonobtuse also for k 3 for this we generalize the class of sub orthocentric tetrahedra that gives rise to the class of ultrametric matrices to sub orthocentric simplices that define symmetric positive definite matrices a with special types of k k principal submatrices for k 3 each sub orthocentric simplices is nonobtuse and we conjecture that any simplex with sub orthocentric facets only is sub orthocentric itself along the way several additional new concepts will be introduced such as vertex gramians simplicial matrix classes the dual hull and the sub orthocentric set of a nonobtuse simplex and nonblocking matrices these concepts may also be of use in a different linear algebraic setting in particular in the context if completely positive matrices for which we will prove some auxiliary results as well

matrix completion problems for pairs of related classes of matrices abstract for a class x of real matrices a list of positions in an n n matrix a pattern is said to have x completion if every partial x matrix that specifies exactly these positions can be completed to an x matrix if x and x0 are classes that satisfy the conditions any partial x matrix is a partial x0 matrix for any x0 matrix a and e 0 a ei is a x matrix and for any partial x matrix a there exists 0 such that a i is a partial x matrix where is the partial identity matrix specifying the same pattern as a then any pattern that has x0 completion must also have x completion however there are usually patterns that have x completion that fail to have x0 completion this result applies to many pairs of subclasses of p and p0 matrices defined by the same restriction on entries including the classes p p0 matrices weakly sign symmetric p p0 matrices and non negative p p0 matrices it also applies to other related pairs of subclasses of p0 matrices such as the pairs classes of p p0 1 matrices weakly sign symmetric p p0 1 matrices and non negative p p0 1 matrices furthermore any pattern that has weakly sign symmetric sign symmetric non negative p0 completion must also have weakly sign symmetric sign symmetric non negative p0 1 completion although these pairs of classes do not satisfy condition 3 similarly the class of inverse m matrices and its topological closure do not satisfy condition 3 but the conclusion remains true and the matrix completion problem for the topological closure of the class of inverse m matrices is solved for patterns containing the diagonal

positive path product and inverse m matrices it is known that an inverse m matrix is strict path product but not necessarily vice versa for n 3 it is shown that any square positive matrix may be made strict path product by predictable additions to the diagonal and that any normalized strict path product matrix may be made inverse m by additions to the diagonal that are bounded in terms of the size of the matrix the latter has implications for pairs of inverse m matrices whose hadamard product is inverse m a determinantal inequality relating principal minors and certain associated almost principal minors is derived for normalized inverse m matrices

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

uniform convergence of a fast algorithm for cauchy singular integral equations abstract error estimates for a fast algorithm applied to a cauchy singular integral equation are proved in weighted uniform norms this algorithm is essentially based on the application of discrete sine transformations numerical results are presented

f divergence inequalities this paper develops systematic approaches to obtain f divergence inequalities dealing with pairs of probability measures defined on arbitrary alphabets functional domination is one such approach where special emphasis is placed on finding the best possible constant upper bounding a ratio of f divergences another approach used for the derivation of bounds among f divergences relies on moment inequalities and the logarithmic convexity property which results in tight bounds on the relative entropy and bhattacharyya distance in terms of chi 2 divergences a rich variety of bounds are shown to hold under boundedness assumptions on the relative information special attention is devoted to the total variation distance and its relation to the relative information and relative entropy including reverse pinsker inequalities as well as on the e gamma divergence which generalizes the total variation distance pinsker s inequality is extended for this type of f divergence a result which leads to an inequality linking the relative entropy and relative information spectrum integral expressions of the renyi divergence in terms of the relative information spectrum are derived leading to bounds on the renyi divergence in terms of either the variational distance or relative entropy

almost principal minors of inverse m matrices abstract it is well known that if an inverse m matrix has a 0 entry then it must be reducible and thus have many more 0 entries this property is actually a special case of a deeper phenomenon that might be loosely described as relations among vanishing almost principal minors in an inverse m matrix this phenomenon encompasses both minors of nested dimension a certain loose monotonicity and minors of the same size in loosely related positions this phenomenon is limited to almost principal minors and where possible converses and examples are given to show the limit of the extent of this phenomenon it is also shown that if one almost principle minor is contained in another then the magnitude of the former is larger than that of the latter

copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints abstract a symmetric matrix c is called copositive if the quadratic form x cx is nonnegative for all nonnegative values of the variables x 1 x 2 x n x a known sufficient condition for a quadratic form x qx to be positive unless x 0 subject to the linear inequality constraints ax 0 is that there should exist a copositive matrix c such that q a ca is positive definite the main result of this paper establishes the necessity of this condition for x qx to be merely nonnegative subject to ax 0 the situation is less straightforward the necessity of the existence of a copositive matrix c such that q a ca is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of a and counterexamples are given to show that in general no such matrix may exist even when slater s constraint qualification holds our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x qx subject to ax 0 in which specific symmetric matrices constructed by rational operations from a and q and depending upon a single real parameter v must be tested for positive definiteness or strict copositivity for large values of v this technique is illustrated by several examples

divergence measures based on the shannon entropy a novel class of information theoretic divergence measures based on the shannon entropy is introduced unlike the well known kullback divergences the new measures do not require the condition of absolute continuity to be satisfied by the probability distributions involved more importantly their close relationship with the variational distance and the probability of misclassification error are established in terms of bounds these bounds are crucial in many applications of divergence measures the measures are also well characterized by the properties of nonnegativity finiteness semiboundedness and boundedness

a polynomial time test for m matrices abstract a real square matrix with nonpositive off diagonal elements is a z matrix and a z matrix with nonnegative principal minors is an m matrix using principal pivoting we derive a polynomial time algorithm for testing whether a z matrix is an m matrix our algorithm is much simpler than a recent one proposed by ramamurthy we also study the zero nonzero structure of the inverse of a nonsingular m matrix and compare a 1 and b 1 in the case that a and b are nonsingular m matrices and b a

on pinsker s and vajda s type inequalities for csiszar s f divergences let d and v denote respectively information divergence and total variation distance pinsker s and vajda s inequalities are respectively d 1 2 v2 and d log 2 v 2 v 2v 2 v in this paper several generalizations and improvements of these inequalities are established for wide classes of f divergences first conditions on f are determined under which an f divergence df will satisfy df cf v2 or df c2 f v2 c4 f v4 where the constants cf c2 f and c4 f are best possible as a consequence lower bounds in terms of v are obtained for many well known distance and divergence measures including the 2 and hellinger s discrimination and the families of tsallis and renyi s divergences for instance if d p q 1 1 p q1 d 1 and p q 1 1 log p q1 d are respectively the relative information of type and the renyi s information gain of order it is shown that d 1 2 v2 1 72 1 2 v4 whenever 1 2 0 1 and that 2 v2 1 36 1 5 5 2 v4 for 0 1 in a somewhat different direction and motivated by the fact that these pinsker s type lower bounds are accurate only for small variation v close to zero lower bounds for df which are accurate for both small and large variation v close to two are also obtained in the special case of the information divergence they imply that d log 2 2 v 2 v 2 log 2 v 2 which uniformly improves vajda s inequality

completions of inverse m matrix patterns abstract a list of positions in an n n real matrix that includes all diagonal positions a pattern is said to have inverse m completion if every partial inverse m matrix that specifies exactly these positions can be completed to an inverse m matrix johnson and smith c r johnson r l smith linear algebra and its applications 241 243 1996 655 667 characterize the positionally symmetric patterns that have inverse m completion as those patterns whose graphs are block clique in this paper the restriction of positional symmetry is removed a pattern has inverse m completion if and only if the digraph g of the pattern has the properties a the induced subdigraph of a cycle is a clique and b if g contains both arc i j and a path of length 1 between i and j then the induced subdigraph of the path is a clique furthermore any irreducible pattern with inverse m completion is positionally symmetric and has a block clique digraph any pattern is permutation similar to a block lower triangular pattern with irreducible diagonal blocks and such a pattern has inverse m completion if and only if i the pattern digraph of each diagonal block is block clique ii each subdiagonal block contains at most one position and iii the pattern digraph of the block structure has no alternate path to a single arc

some results on structured m matrices with an application to wireless communications in this paper we study conditions under which a specially structured z matrix is an m matrix we apply the result to a capacity problem in wireless communications

a new metric for probability distributions we introduce a metric for probability distributions which is bounded information theoretically motivated and has a natural bayesian interpretation the square root of the well known spl chi sup 2 distance is an asymptotic approximation to it moreover it is a close relative of the capacitory discrimination and jensen shannon divergence

on the convexity of some divergence measures based on entropy functions three measures of divergence between vectors in a convex set of a n dimensional real vector space are defined in terms of certain types of entropy functions and their convexity property is studied among other results a classification of the entropies of degree alpha is obtained by the convexity of these measures these results have applications in information theory and biological studies

theorems on schur complement of block diagonally dominant matrices and their application in reducing the order for the solution of large scale linear systems abstract we firstly consider the block dominant degree for i ii block strictly diagonally dominant matrix and their schur complements showing that the block dominant degree for the schur complement of an i ii block strictly diagonally dominant matrix is greater than that of the original grand block matrix then as application we present some disc theorems and some bounds for the eigenvalues of the schur complement by the elements of the original matrix further by means of matrix partition and the schur complement of block matrix based on the derived disc theorems we give a kind of iteration called the schur based iteration which can solve large scale linear systems though reducing the order by the schur complement and the numerical example illustrates that the iteration can compute out the results faster

on maximum principles for monotone matrices abstract for monotone matrices and h matrices maximum principles are defined and necessary and sufficient conditions are derived for these principles to hold

on the spectra of matrices having nonnegative sums of principal minors abstract the localization of the eigenvalues of matrices with nonnegative sums of principal minors is studied the discussion is carried out in particular for p matrices such matrices for which almost all the eigenvalues lie on the left half plane are constructed

notes on f0 matrices abstract the author studies f 0 matrices and shows that n n matrix a f 0 if and only if a ti b with b 0 and n 2 b t n 1 b where k b denotes the maximum of the spectral radii of all k principal submatrices of b also an inverse f 0 matrix problem is considered

refinements of pinsker s inequality let v and d denote respectively total variation and divergence we study lower bounds of d with v fixed the theoretically best i e largest lower bound determines a function l l v vajda s 1970 tight lower bound the main result is an exact parametrization of l this leads to taylor polynomials which are lower bounds for l and thereby to extensions of the classical pinsker 1960 inequality which has numerous applications cf pinsker and followers