bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

quantitative uniqueness of elliptic equations based on a variant of frequency function we improve the vanishing order of solutions for schr o dinger equations which describes quantitative behavior of strong uniqueness continuation property for the first time we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions furthermore strong unique continuation is established for higher order elliptic equations using this variant of frequency function

fiedler companion linearizations for rectangular matrix polynomials the development of new classes of linearizations of square matrix polynomials that generalize the classical first and second frobenius companion forms has attracted much attention in the last decade research in this area has two main goals finding linearizations that retain whatever structure the original polynomial might possess and improving properties that are essential for accurate numerical computation such as eigenvalue condition numbers and backward errors however all recent progress on linearizations has been restricted to square matrix polynomials since rectangular polynomials arise in many applications it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials in this paper the family of fiedler linearizations is extended from square to rectangular matrix polynomials and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials fiedler linearizations are one of the most important classes of linearizations introduced in recent years but their generalization to rectangular polynomials is nontrivial and requires a completely different approach to the one used in the square case to the best of our knowledge this is the first class of new linearizations that has been generalized to rectangular polynomials

a note on permanents and generalized complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices 3 the multiplication of the g i s is intrinsic connections between the 2 2 matrices c k and the resulting cb matrix g k have been explored in particular the properties which are inherited from the c k to the g k two situations were considered for the ordinary real cb matrices and for the corresponding sign pattern matrices in this paper we consider generalized complementary basic matrices where the c k matrices are replaced by square matrices of arbitrary sizes some interesting facts and results about the permanents and permanental polynomials of such matrices are obtained

structure ranks of matrices abstract the structure rank of a matrix i e the maximum order of a nonsingular submatrix all of whose entries are located in a given structure a subset of m n where m n are the row and column index sets respectively were previously studied for the off diagonal part here results on more general ranks e g off block diagonal strictly upper triangular and strictly upper block triangular are presented in particular it is shown that for a nonsingular matrix a in each of the last mentioned three cases the ranks of a and a 1 coincide

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

ray patterns of matrices and nonsingularity a complex matrix a is ray nonsingular if det x 0 a f 0 for every matrix x with positive entries a sufficient condition for ray nonsingularity is that the origin is not in the relative interior of the convex hull of the signed transversal products of a the concept of an isolated set of transversals is defined and used to obtain a neces

sharp vanishing order of solutions to stationary schrodinger equations on carnot groups of arbitrary step abstract based on a variant of the frequency function approach of almgren 1 under appropriate assumptions we establish an optimal upper bound on the vanishing order of solutions to stationary schrodinger equations associated to sub laplacian on carnot groups of arbitrary step such a bound provides a quantitative form of strong unique continuation and can be thought of as a subelliptic analogue of the recent results obtained by bakri 3 and zhu 27 for the standard laplacian

factorization of hessenberg matrices abstract the analysis of any hessenberg matrix as a product of a companion matrix and a triangular matrix is presented in this paper the factors are explicitly given on terms of the entries of the hessenberg matrix and further various factorizations are undertaken applications to comrade and confederate matrices as well as for the corresponding block cases are included

comments on adaptive manipulator control a case study by j slotine and w li the commenter establishes the stability in the sense of lyapunov of the adaptive motion controller for robot manipulators reported by j slotine and w li ibid vol 33 p 995 1003 nov 1988

characterization of a family of generalized companion matrices abstract matrices a of order n having entries in the field f x 1 x n of rational functions over a field f and characteristic polynomial det t i a t n x 1 t n 1 x n 1 t x n are studied it is known that such matrices are irreducible and have at least 2 n 1 nonzero entries such matrices with exactly 2 n 1 nonzero entries are called ma zhan matrices conditions are given that imply that a ma zhan matrix is similar via a monomial matrix to a generalized companion matrix that is a lower hessenberg matrix with ones on its superdiagonal and exactly one nonzero entry in each of its subdiagonals via the ax grothendieck theorem respectively its analog for the reals these conditions are shown to hold for a family of matrices whose entries are complex respectively real polynomials

max algebraic complementary basic matrices abstract a max algebraic version of the results on complementary basic matrices is presented it is shown that the max permanent of the result is equal to the product of simpler max permanents and the finite max eigenvalues of the product are the same for any permutation of the basic matrices

quantitative uniqueness of some higher order elliptic equations abstract we study the quantitative unique continuation property of some higher order elliptic operators we prove a lower bound for nontrivial solutions of the equation m u v x u 0 m n v x is bounded the bound shows that nontrivial solutions can not decay faster than e x 4 3 ln x at infinity moreover we obtain an improved lower bound of nontrivial solutions for a special forth order elliptic operators in dimension 2 the bound is shown to be essentially sharp by constructing a meshkov type example

adaptive control of robot manipulator using fuzzy compensator this paper presents two kinds of adaptive control schemes for robot manipulator which has the parametric uncertainties in order to compensate these uncertainties we use the fls fuzzy logic system that has the capability to approximate any nonlinear function over the compact input space in the proposed control schemes we need not derive the linear formulation of robot dynamic equation and tune the parameters we also suggest the robust adaptive control laws in all proposed schemes for decreasing the effect of approximation error to reduce the number of fuzzy rules of the fls we consider the properties of robot dynamics and the decomposition of the uncertainty function the proposed controllers are robust not only to the structured uncertainty such as payload parameter but also to the unstructured one such as friction model and disturbance the validity of the control scheme is shown by computer simulations of a two link planar robot manipulator

root polynomials and their role in the theory of matrix polynomials abstract we develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials i e either regular or singular and study how these vector polynomials are related to the spectral properties of matrix polynomials we pay particular attention to the so called maximal sets of root polynomials and prove that they carry complete information about the eigenvalues finite or infinite of matrix polynomials and that they are related to the matrices that transform any matrix polynomial into its smith form in addition we describe clearly for the first time in the literature the extremality properties of such maximal sets and identify some of them whose vectors have minimal grade once the main theoretical properties of root polynomials have been established the interaction of root polynomials with three problems that have attracted considerable attention in the literature is analyzed more precisely we study the change of root polynomials under rational transformations or reparametrizations of matrix polynomials the recovery of the root polynomials of a matrix polynomial from those of its most important linearizations and the relationship between the root polynomials of two dual pencils we emphasize that for the case of regular matrix polynomials all the results in this paper can be translated into the language of jordan chains as a consequence of the well known relationship between root polynomials and jordan chains therefore a number of open problems are also solved for jordan chains of regular matrix polynomials we also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials

on the permanents of circulant and degenerate schur matrices abstract we communicate three formulas for the permanents of degenerate schur and circulant matrices these combinatorial and integral formulas are intended for the analytical and asymptotic evaluation of the permanents as well as for the solution of three dimensional ising model the paper s goal is to draw attention to the open fundamental problem of finding the permanents asymptotics a solution to this problem would be tremendously important for physics of many body systems and critical phenomena as well as for quantum field theory

structured strong linearizations from fiedler pencils with repetition ii abstract in this paper we give strong linearizations of a matrix polynomial p preserving the skew symmetry or t alternating structure of p the linearizations obtained are of the form s l where l is a block symmetric fiedler pencil with repetition and s is a direct sum of blocks of the form i or i with i the identity matrix this paper is a continuation of 2 where the corresponding problem for p with a symmetric structure was studied and as a consequence the block symmetric fiedler pencils with repetition were characterized

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

on the fractional landis conjecture abstract in this paper we study a landis type conjecture for fractional schrodinger equations of fractional power s 0 1 with potentials we discuss both the cases of differentiable and non differentiable potentials on the one hand it turns out for differentiable potentials with some a priori bounds if a solution decays at a rate e x 1 then this solution is trivial on the other hand for s 1 4 1 and merely bounded non differentiable potentials if a solution decays at a rate e x with 4 s 4 s 1 then this solution must again be trivial remark that when s 1 4 s 4 s 1 4 3 which is the optimal exponent for the standard laplacian for the case of non differentiable potentials and s 1 4 1 we also derive a quantitative estimate mimicking the classical result by bourgain and kenig

dominant matrices and max algebra abstract we study the class of so called totally dominant matrices in the usual algebra and in the max algebra in which the sum is the maximum and the multiplication is usual it turns out that this class coincides with the well known class of positive matrices having positive the determinants of all 2 2 submatrices the closure of this class is closed not only with respect to the usual but also with respect to the max multiplication further properties analogous to those of totally positive matrices are proved and some connections to monge matrices are mentioned

modified gershgorin disks for companion matrices all the zeros of a polynomial are contained in the union of gershgorin disks derived from its companion matrix a consequence of gershgorin s theorem however this theorem does not exploit the structure of the companion matrix we will use this structure to obtain smaller zero inclusion regions thereby providing some nonstandard results to accompany and illustrate this frequently covered topic in numerical and matrix analysis

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

subtotally positive and monge matrices abstract a real matrix is called k subtotally positive if the determinants of all its submatrices of order at most k are positive we show that for an m n matrix only mn inequalities determine such class for every k 1 k min m n spectral properties of square k subtotally positive matrices are studied finally completion problems for 2 subtotally positive matrices and their additive counterpart the anti monge matrices are investigated since totally positive matrices are 2 subtotally positive as well the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices

adaptive control based on explicit model of robot manipulator adaptive and nonadaptive control algorithms which make use of a fundamental mathematical property concerning positive definite matrices and lyapunov stability theory are proposed for the control of robot manipulators using the fact that the matrix dd q dt 2c q dq dt is skew symmetric nonadaptive controllers which have a simplified structure with less computational burden are proposed using the dynamic equations for robot manipulators parameter adaptation rules are developed for updating the controller s partially or totally unknown parameters generalizing them to model reference adaptive controllers to further take advantage of the simplified structure of the proposed adaptive controllers a method for deriving the dynamic model of a robot manipulator which is linear in terms of its parameters is given this dynamic model is also suitable for the pure identification of the parameters of links and payload of the manipulator

a simplified approach to fiedler like pencils via block minimal bases pencils abstract the standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix coefficients of the polynomial into a matrix pencil transforming the problem into an equivalent generalized eigenvalue problem such pencils are known as linearizations many of the families of linearizations for matrix polynomials available in the literature are extensions of the so called family of fiedler pencils these families are known as generalized fiedler pencils fiedler pencils with repetition and generalized fiedler pencils with repetition or fiedler like pencils for simplicity the goal of this work is to unify the fiedler like pencils approach with the more recent one based on strong block minimal bases pencils introduced in f m dopico et al 2017 17 to this end we introduce a family of pencils that we have named extended block kronecker pencils whose members are under some generic nonsingularity conditions strong block minimal bases pencils and show that with the exception of the non proper generalized fiedler pencils all fiedler like pencils belong to this family modulo permutations as a consequence of this result we obtain a much simpler theory for fiedler like pencils than the one available so far moreover we expect this simplification to allow for further developments in the theory of fiedler like pencils such as global or local backward error analyses and eigenvalue conditioning analyses of polynomial eigenvalue problems solved via fiedler like linearizations

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

dense alternating sign matrices and extensions abstract in this paper some connections of dense alternating sign matrices with total unimodularity combined matrices and generalized complementary basic matrices are explored

on adaptive inverse dynamics control of rigid robots the authors derive an adaptive inverse dynamics control law for rigid robots that overcomes the most restrictive difficulty of such schemes to date namely the requirement that the inverse of the estimated inertia matrix remain bounded the control scheme still requires the joint accelerations for its implementation in practice one would implement this scheme by estimating the acceleration from the measured velocity good performance could still be expected without direct acceleration measurement assuming sufficient encoder resolution and sufficiently fast sampling another practical advantage of the authors scheme is that since the nominal inverse dynamics or inner loop portion of their control law is fixed it is more amenable to the development of a dedicated interface and can be burned onto a chip to reduce online computation requirements

adaptive fuzzy sliding mode control of nonlinear system in this paper the fuzzy approximator and sliding mode control smc scheme are considered we propose two methods of adaptive smc schemes that the fuzzy logic systems approximators are used to approximate the unknown system functions in designing the smc of nonlinear system in the first method a fuzzy logic system is utilized to approximate the unknown function f of the nonlinear system x sup n f x t b x t u and the robust adaptive law is proposed to reduce the approximation errors between the true nonlinear functions and fuzzy approximators in the second method two fuzzy logic systems are utilized to approximate the f and b respectively and the control law which is robust to approximation error is also designed the stabilities of proposed control schemes are proved and these schemes are applied to an inverted pendulum system the comparisons between the proposed control schemes are shown in simulations

a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

green s matrices abstract our goal is to identify and understand matrices a that share essential properties of the unitary hessenberg matrices m that are fundamental for szego s orthogonal polynomials those properties include i recurrence relations connect characteristic polynomials rk x of principal minors of a ii a is determined by generators parameters generalizing reflection coefficients of unitary hessenberg theory iii polynomials rk x correspond not only to a but also to a certain cmv like five diagonal matrix iv the five diagonal matrix factors into a product bc of block diagonal matrices with 2 2 blocks v submatrices above and below the main diagonal of a have rank 1 vi a is a multiplication operator in the appropriate basis of laurent polynomials vii eigenvectors of a can be expressed in terms of those polynomials conditions v connects our analysis to the study of quasi separable matrices but the factorization requirement iv narrows it to the subclass of green s matrices that share properties i vii the key tool is twist transformations that provide 2n matrices all sharing characteristic polynomials of principal minors with a one such twist transformation connects unitary hessenberg to cmv another twist transformation explains findings of fiedler who noticed that companion matrices give examples outside the unitary hessenberg framework we mention briefly the further example of a daubechies wavelet matrix infinite matrices are included

integrally normalizable matrices and zero nonzero patterns abstract the problem of determining whether or not an n n integer matrix is diagonally similar to an integer matrix with n 1 off diagonal entries equal to 1 is studied such a matrix is called integrally normalizable and a zero nonzero pattern is integrally normalizable if each matrix with this zero nonzero pattern is integrally normalizable with respect to the same set of n 1 off diagonal entries matrices that are integrally normalizable with respect to a fixed spanning tree and integrally normalizable zero nonzero patterns are characterized the maximum number of nonzero entries in an n n integrally normalizable zero nonzero pattern is shown to be n 2 3 n 2 2 extensions to matrices over other integral domains are also presented

companion factorization in the general linear group gl n c and applications abstract the companion factorization for nonsingular matrices belonging to the general linear group gl n c is studied here the entries in the last row of the companion matrices are explicitly represented in terms of determinants of proper sub matrices of the matrix being factorized the computation of the inverse of a nonsingular matrix is given as an application of this factorization

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

three circle theorems for eigenfunctions on complete shrinking gradient ricci solitons with constant scalar curvature abstract the classical hadamard s three circle theorem is generalized to complete shrinking gradient ricci solitons with constant scalar curvature for eigenfunctions of the laplace operator based on a variant of frequency function

block minimal bases l ifications of matrix polynomials abstract the standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil known as a strong linearization this process transforms the problem into an equivalent generalized eigenvalue problem however there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials this has motivated the idea of a strong l ification of a matrix polynomial which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial we present in this work a novel method for constructing strong l ifications of matrix polynomials of size m n and grade d when l d and l divides nd or md this method is based on a family called strong block minimal bases matrix polynomials and relies heavily on properties of dual minimal bases we show how strong block minimal bases l ifications can be constructed from the coefficients of a given matrix polynomial p we also show that these l ifications satisfy many desirable properties for numerical applications they are strong l ifications regardless of whether p is regular or singular the minimal indices of the l ifications are related to those of p via constant uniform shifts and eigenvectors and minimal bases of p can be recovered from those of any of the strong block minimal bases l ifications in the special case where l divides d we introduce a subfamily of strong block minimal bases matrix polynomials named block kronecker matrix polynomials which is shown to be a fruitful source of companion l ifications

block kronecker linearizations of matrix polynomials and their backward errors we introduce a new family of strong linearizations of matrix polynomials which we call block kronecker pencils and perform a backward stability analysis of complete polynomial eigenproblems these problems are solved by applying any backward stable algorithm to a block kronecker pencil such as the staircase algorithm for singular pencils or the qz algorithm for regular pencils this analysis allows us to identify those block kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial these favorable pencils include the famous fiedler linearizations which are just a very particular case of block kronecker pencils thus our analysis offers the first proof available in the literature of global backward stability for fiedler pencils in addition the theory developed for block kronecker pencils is much simpler than the theory available for fiedler pencils especially in the case of rectangular matrix polynomials the global backward error analysis in this work presents for the first time the following key properties it is a rigorous analysis valid for finite perturbations i e it is not a first order analysis it provides precise bounds it is valid simultaneously for a large class of linearizations and it establishes a framework that may be generalized to other classes of linearizations these features are related to the fact that block kronecker pencils are a particular case of the new family of strong block minimal bases pencils which include certain perturbations of block kronecker pencils this allows us to extend the results in this paper to other contexts

companion matrix patterns abstract companion matrices especially the frobenius companion matrices are used in algorithms for finding roots of polynomials and are also used to find bounds on eigenvalues of matrices in 2003 fiedler introduced a larger class of companion matrices that includes the frobenius companion matrices one property of the combinatorial pattern of these companion matrices is that up to diagonal similarity they uniquely realize every possible spectrum of a real matrix we characterize matrix patterns that have this property and consequently introduce more companion matrix patterns we observe that each fiedler companion matrix is permutationally similar to a unit hessenberg matrix we provide digraph characterizations of the classes of patterns described and in particular all sparse companion matrices noting that there are companion matrices that are not sparse

complementary basic matrices abstract we show that an n n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions except the first and last completely between the subdiagonal and superdiagonal part then this matrix can be factorized into a product of n 1 matrices each consisting of a 2 2 principal submatrix in two consecutive rows and columns in all possible of the n 1 positions and completed by ones along the diagonal the converse is also true it is shown that the spectrum does not depend on the order of the factors

quantitative uniqueness of solutions to parabolic equations abstract we investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth riemannian manifolds quantitative uniqueness is a quantitative form of strong unique continuation property we characterize quantitative uniqueness by the rate of vanishing we can obtain the sharp vanishing order for solutions in term of the c 1 1 norm of the potential functions as well as the l norm of the coefficient functions some new quantitative carleman estimates and three cylinder inequalities are established

intrinsic products and factorizations of matrices we say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition we present several examples together with important applications these applications include companion matrices and sign nonsingular matrices

condition numbers for inversion of fiedler companion matrices abstract the fiedler matrices of a monic polynomial p z of degree n are n n matrices with characteristic polynomial equal to p z and whose nonzero entries are either 1 or minus the coefficients of p z fiedler matrices include as particular cases the classical frobenius companion forms of p z frobenius companion matrices appear frequently in the literature on control and signal processing but it is well known that they posses many properties that are undesirable numerically which limit their use in applications in particular as n increases frobenius companion matrices are often nearly singular i e their condition numbers for inversion are very large therefore it is natural to investigate whether other fiedler matrices are better conditioned than the frobenius companion matrices or not in this paper we present explicit expressions for the condition numbers for inversion of all fiedler matrices with respect the frobenius norm i e a f ij a ij 2 this allows us to get a very simple criterion for ordering all fiedler matrices according to increasing condition numbers and to provide lower and upper bounds on the ratio of the condition numbers of any pair of fiedler matrices these results establish that if p 0 1 then the frobenius companion matrices have the largest condition number among all fiedler matrices of p z and that if p 0 1 then the frobenius companion matrices have the smallest condition number we also provide families of polynomials where the ratio of the condition numbers of pairs of fiedler matrices can be arbitrarily large and prove that this can only happen when both fiedler matrices are very ill conditioned we finally study some properties of the singular values of fiedler matrices and determine how many of the singular values of a fiedler matrix are equal to one

some inheritance properties for complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices m fiedler a note on companion matrices linear algebra appl 372 2003 325 331 9 3 the multiplication of the g i s is intrinsic we explore connections between the 2 2 matrices c k and the resulting cb matrix g k in particular which properties are inherited from the c k to the g k we consider two situations for the ordinary real cb matrices and for the corresponding sign pattern matrices

polynomial zigzag matrices dual minimal bases and the realization of completely singular polynomials minimal bases of rational vector spaces are a well known and important tool in systems theory if minimal bases for two subspaces of rational n space are displayed as the rows of polynomial matrices z 1 lambda k times n and z 2 lambda m times n respectively then z 1 and z 2 are said to be dual minimal bases if the subspaces have complementary dimension i e k m n and z 1 lambda z 2 t lambda 0 in other words each z j lambda provides a minimal basis for the nullspace of the other it has long been known that for any dual minimal bases z 1 lambda and z 2 lambda the row degree sums of z 1 and z 2 are the same in this paper we show that this is the only constraint on the row degrees thus characterizing the possible row degrees of dual minimal bases the proof is constructive making extensive use of a new class of sparse structured polynomial matrices that we have baptized zigzag matrices another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices given a list of left and right minimal indices and a desired degree d does there exist a completely singular matrix polynomial i e a matrix polynomial with no elementary divisors whatsoever of degree d having exactly the prescribed minimal indices we show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices the constructed realization is simple and explicitly displays the desired minimal indices in a fashion analogous to the classical kronecker canonical form of singular pencils

reprint of a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

the continuing influence of fiedler s work on companion matrices this is a reconstruction in article like form of a talk given at the minisymposium in honor of miroslav fiedler at the 17th ilas conference held at tu braunschweig germany on thurs 25 aug 2011

counting fiedler pencils with repetitions abstract we introduce a new notation based on diagrams to deal with fiedler pencils with repetitions fpr and use it to solve several counting problems in particular we give explicit recurrences to count the number of fprs of a given degree d the number of symmetric palindromic and antipalindromic ones where the latter two structures are intended in the sense of 5 we relate these structures to the presence of symmetries in the associated diagrams

some graph theoretic properties of generalized complementary basic matrices abstract this paper investigates graph theoretic properties of the generalized complementary basic matrices these include cyclic properties hamiltonian properties block connected properties and minimal strongly connected digraph properties

structured strong linearizations from fiedler pencils with repetition i abstract in many applications the polynomial eigenvalue problem p x 0 arises with p being a structured matrix polynomial for example palindromic symmetric skew symmetric in order to solve a structured polynomial eigenvalue problem it is convenient to use strong linearizations with the same structure as p to ensure that the symmetries in the eigenvalues due to that structure are preserved in numerical computations in this paper we characterize all the pencils in the family of the fiedler pencils with repetition introduced by vologiannidis and antoniou 25 associated with a square matrix polynomial p that are block symmetric for every matrix polynomial p we show that this family of pencils is precisely the set of all fiedler pencils with repetition that are symmetric when p is when some generic nonsingularity conditions are satisfied these pencils are strong linearizations of p in particular our family strictly contains the standard basis for dl p a k dimensional vector space of symmetric pencils introduced by mackey mackey mehl and mehrmann 20

generalized totally positive matrices abstract we say that a rectangular matrix over a in general noncommutative ring with identity having a positive part is generalized totally positive gtp if in all nested sequences of so called relevant submatrices the schur complements are positive here a relevant submatrix is such either having k consecutive rows and the first k columns or k consecutive columns and the first k rows this notion generalizes the usual totally positive matrices we prove e g that a square matrix is gtp if and only if it admits a certain factorization with bidiagonal type factors and certain invertible entries also the product of square gtp matrices of the same order is again a gtp matrix and its inverse has the checkerboard sign property

totally nonnegative 0 1 matrices abstract we investigate 0 1 matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers such matrices are characterized by four forbidden submatrices of orders 2 and 3 we show that the maximum number of 0 s in an irreducible totally nonnegative 0 1 matrix of order n is n 1 2 and characterize those matrices with this number of 0 s we also show that the minimum perron value of an irreducible totally nonnegative 0 1 matrix of order n equals 2 2 cos 2 n 2 and characterize those matrices with this perron value

linearizations of matrix polynomials in newton bases we discuss matrix polynomials expressed in a newton basis and the associated polynomial eigenvalue problems properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces also we show how the family of fiedler pencils can be adapted to matrix polynomials expressed in a newton basis these new newton fiedler pencils are shown to be strong linearizations and some computational aspects related to them are discussed

max algebra the linear algebra of combinatorics abstract let a b max a b a b a b for a b r r by max algebra we understand the analogue of linear algebra developed for the pair of operations extended to matrices and vectors max algebra which has been studied for more than 40 years is an attractive way of describing a class of nonlinear problems appearing for instance in machine scheduling information technology and discrete event dynamic systems this paper focuses on presenting a number of links between basic max algebraic problems like systems of linear equations eigenvalue eigenvector problem linear independence regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand this indicates that max algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems the paper is intended for wider readership including researchers not familiar with max algebra

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m