miniversal deformations of pairs of skew symmetric matrices under congruence for each pair of complex symmetric matrices a b we provide a normal form with a minimal number of independent parameters to which all pairs of complex symmetric matrices a over tilde b ove

on vector spaces of linearizations for matrix polynomials in orthogonal bases abstract regular and singular matrix polynomials p i 0 k p i i p i r n n given in an orthogonal basis 0 1 k are considered following the ideas in 9 the vector spaces called m 1 p m 2 p and dm p of potential linearizations for p are analyzed all pencils in m 1 p are characterized concisely moreover several easy to check criteria whether a pencil in m 1 p is a strong linearization of p are given the equivalence of some of them to the z rank condition 9 is pointed out results on the vector space dimensions the genericity of linearizations in m 1 p and the form of block symmetric pencils are derived in a new way on a basic algebraic level moreover an extension of these results to degree graded bases is presented throughout the paper structural resemblances between the matrix pencils in l 1 i e the results obtained in 9 and their generalized versions are pointed out

an algorithm for exact eigenvalue calculations for rotationally periodic structures an existing algorithm ensures that no eigenvalues are missed when using the stiffness matrix method of structural analysis where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems the algorithm permits efficient multi level substructuring and gives exact results when the member equations used are those obtained by solving appropriate differential equations the present paper extends this algorithm to cover rotationally periodic i e cyclically symmetric three dimensional structures which are analysed by using complex arithmetic to obtain a stiffness matrix which involves only one of the rotationally repeating portions of the structure nodes and members are allowed to coincide with the axis of rotational periodicity and the resulting modes are classified rigid body freedoms are accounted for empirically and the exact member equations and efficient multi level substructuring of the earlier algorithm can be used when assembling the stiffness matrix of the repeating portion

generic skew symmetric matrix polynomials with fixed rank and fixed odd grade we show that the set of m m complex skew symmetric matrix polynomials of odd grade d i e of degree at most d and normal rank at most 2r is the closure of the single set of matrix polynomials

the impact of a systematic reduction in shoe floor friction on heel contact walking kinematics a gait simulation approach falls initiated by slips and trips are a serious health hazard to older adults experimental studies have provided important descriptions of postural responses to slipping but it is difficult to determine why some slips result in falls from experiments alone computational modeling and simulation techniques can complement experimental approaches by identifying causes of failed recovery attempts the purpose of this study was to develop a method to determine the impact of a systematic reduction in the foot floor friction coefficient mu on the kinematics of walking shortly after heel contact approximately 200ms a walking model that included foot floor interactions was utilized to find the set of moments that best tracked the joint angles and measured ground reaction forces obtained from a non slipping dry trial a passive slip was simulated by driving the model with the joint moments from the dry simulation and by reducing mu slip simulations with values of mu greater than the subject specific peak required coefficient of friction rcof an experimental measure of slip resistant gait resulted in only minor deviations in gait kinematics from the dry condition in contrast slip simulations run in environments characterized by mu language en

on the inverse eigenvalue problem for t alternating and t palindromic matrix polynomials abstract the inverse eigenvalue problem for t alternating matrix polynomials over arbitrary algebraically closed fields of characteristic different from two is considered the main result shows that the necessary conditions obtained in 9 for a matrix polynomial to be the smith form of a t alternating matrix polynomial are under mild conditions also sufficient to be the smith form of a t alternating matrix polynomial with invertible leading coefficient which is additionally in anti triangular form in particular this result implies that any t alternating matrix polynomial with invertible leading coefficient is equivalent to a t alternating matrix polynomial in anti triangular form that has the same finite and infinite elementary divisors as the original matrix polynomial finally the inverse eigenvalue problem for t palindromic matrix polynomials is considered excluding the case that both 1 and 1 are eigenvalues

inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems abstract sylvester s law of inertia states that the number of positive negative and zero eigenvalues of hermitian matrices is preserved under congruence transformations the same is true of generalized hermitian definite eigenvalue problems in which the two matrices are allowed to undergo different congruence transformations but not for the indefinite case in this paper we investigate the possible change in inertia under congruence for generalized hermitian indefinite eigenproblems and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil especially when one of the matrices is almost definite a prominent application of the original sylvester s law is in finding the number of eigenvalues in an interval our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

jordan structures of alternating matrix polynomials alternating matrix polynomials that is polynomials whose coefficients alternate between symmetric and skew symmetric matrices generalize the notions of even and odd scalar polynomials we investigate the smith forms of alternating matrix polynomials showing that each invariant factor is an even or odd scalar polynomial necessary and sufficient conditions are derived for a given smith form to be that of an alternating matrix polynomial these conditions allow a characterization of the possible jordan structures of alternating matrix polynomials and also lead to necessary and sufficient conditions for the existence of structure preserving strong linearizations most of the results are applicable to singular as well as regular matrix polynomials

direct complements of invariant lagrangian subspaces and minimal factorizations of skew symmetric rational matrix functions abstract symmetric factorizations of skew symmetric real rational matrix functions are studied in particular it is proved that generically a proper skew symmetric real rational matrix mxm matrix function w w t with w invertible can be factorized minimally as w l tdl for a real rational mxm matrix function l applications are given concerning symmetric solutions of matrix quadratic equations

block diagonalization of adjacency and laplacian matrices for graph product applications in structural mechanics eigenvalues and eigenvectors of graphs have many applications in structural mechanics and combinatorial optimization for a regular space structure the visualization of its graph model as the product of two simple graphs results in a substantial simplification in the solution of the corresponding eigenproblems in this paper the adjacency and laplacian matrices of four graph products namely cartesian strong cartesian direct and lexicographic products are diagonalized and efficient methods are obtained for calculating their eigenvalues and eigenvectors an exceptionally efficient method is developed for the eigensolution of the laplacian matrices of strong cartesian and direct products special attention is paid to the lexicographic product which is not studied in the past as extensively as the other three graph products examples are provided to illustrate some applications of the methods in structural mechanics copyright 2006 john wiley sons ltd

quadratic realizability of palindromic matrix polynomials let cl cl 1 cl 2 be a list consisting of a sublist cl 1 of powers of irreducible monic scalar polynomials over an algebraically closed field ff and a sublist cl 2 of nonnegative integers for an arbitrary such list cl we give easily verifiable necessary and sufficient conditions for cl to be the list of elementary divisors and minimal indices of some t palindromic quadratic matrix polynomial with entries in the field ff for cl satisfying these conditions we show how to explicitly construct a t palindromic quadratic matrix polynomial having cl as its structural data that is we provide a t palindromic quadratic realization of cl our construction of t palindromic realizations is accomplished by taking a direct sum of low bandwidth t palindromic blocks closely resembling the kronecker canonical form of matrix pencils an immediate consequence of our in depth study of the structure of t palindromic quadratic polynomials is that all even grade t palindromic matrix polynomials have a t palindromic strong quadratification finally using a particular m o bius transformation we show how all of our results can be easily extended to quadratic matrix polynomials with t even structure

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

an efficient analysis of repetitive structures generated by graph products in this paper an efficient method is developed for the analysis of regular structures a structure is called regular if its model can be formed by a graph product here instead of direct solution of the equations corresponding to a regular structure or finding the inverse of the stiffness matrix directly modal analysis is used and eigenvectors are employed for calculating the displacements and then internal forces of the structures for this purpose first an efficient method is developed for calculating the eigenvectors of the product graphs and then a method is presented for using these eigenvectors for evaluating the displacements of a structure

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

hamiltonian square roots of skew hamiltonian matrices abstract we present a constructive existence proof that every real skew hamiltonian matrix w has a real hamiltonian square root the key step in this construction shows how one may bring any such w into a real quasi jordan canonical form via symplectic similarity we show further that every w has infinitely many real hamiltonian square roots and give a lower bound on the dimension of the set of all such square roots some extensions to complex matrices are also presented

response time is more important than walking speed for the ability of older adults to avoid a fall after a trip abstract we previously reported that the probability of an older adult recovering from a forward trip and using a lowering strategy increases with decreased walking velocity and faster response time to determine the within subject interaction of these variables we asked three questions 1 is the body orientation at the time that the recovery foot is lowered to the ground tilt angle critical for successful recovery 2 can a simple inverted pendulum model using subject specific walking velocity and response time as input variables predict this body orientation and thus success of recovery 3 is slower walking velocity or faster response time more effective in preventing a fall after a trip tilt angle was a perfect predictor of a successful recovery step indicating that the recovery foot placement must occur before the tilt angle exceeds a critical value of between 23 and 26 from vertical the inverted pendulum model predicted the tilt angle from walking velocity and response time with an error of 0 4 2 2 and a correlation coefficient of 0 93 the model predicted that faster response time was more important than slower walking velocity for successful recovery in a typical individual who is at risk for falling we predicted that a reduction of response time to a normal value allows a 77 increase in safe walking velocity the mathematical model produced patient specific recommendations for fall prevention and suggested the importance of directing therapeutic interventions toward improving the response time of older adults

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

structure preserving stratification of skew symmetric matrix polynomials we study how elementary divisors and minimal indices of a skew symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients we investigate these change

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

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

exact eigenvalue calculations for structures with rotationally periodic substructures on considere 2 classes de sous structures a symetrie cyclique autour d un axe longitudinal l une n est reliee a sa structure mere qu aux extremites de son axe tandis que l autre est reliee par un nombre quelconque de n uds a sa structure mere qui partage alors sa periodicite rotationnelle

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

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

palindromic companion forms for matrix polynomials of odd degree the standard way to solve polynomial eigenvalue problems p x 0 is to convert the matrix polynomial p into a matrix pencil that preserves its spectral information a process known as linearization when p is palindromic the eigenvalues elementary divisors and minimal indices of p have certain symmetries that can be lost when using the classical first and second frobenius companion linearizations for numerical computations since these linearizations do not preserve the palindromic structure recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original p might possess with particular attention to the preservation of palindromic structure however no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved in this paper we present a family of linearizations for odd degree polynomials p which are palindromic whenever p is and which are valid for all palindromic polynomials of odd degree we illustrate our construction with several examples in addition we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family

on computing the eigenvalues of a symplectic pencil this paper presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete time algebraic riccati equation the algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil it requires about one fourth the number of floating point operations that the qz algorithm uses to compute the eigenvalues of the pencil directly the proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by van loan for computing the eigenvalues of hamiltonian matrices

on the sign characteristic of hermitian linearizations in dl p abstract the computation of eigenvalues and eigenvectors of matrix polynomials is an important but difficult problem the standard approach to solve this problem is to use linearizations which are matrix polynomials of degree 1 that share the eigenvalues of p hermitian matrix polynomials and their real eigenvalues are of particular interest in applications attached to these eigenvalues is a set of signs called the sign characteristic from both a theoretical and a practical point of view it is important to be able to recover the sign characteristic of a hermitian linearization of p from the sign characteristic of p in this paper for a hermitian matrix polynomial p with nonsingular leading coefficient we describe in terms of the sign characteristic of p the sign characteristic of the hermitian linearizations in the vector space dl p mackey mackey mehl and mehrmann 2006 in particular we identify the hermitian linearizations in dl p that preserve the sign characteristic of p we also provide a description of the sign characteristic of the hermitian linearizations of p in the family of generalized fiedler pencils with repetition bueno dopico furtado and rychnovsky 2015

minimal step length necessary for recovery of forward balance loss with a single step although the boundary conditions necessary to trigger a step in reaction to a forward balance loss have been predicted in previous research the relationship between minimal step length needed for balance recovery with this single step and the center of mass com motion state i e its position and velocity remains unknown the purpose of this paper was to present a theoretical framework within which the minimal step length needed for balance recovery can be estimated we therefore developed a simplified four segment sagittal model of human body stepping for balance recovery the work energy principle of the newtonian mechanics was employed in the simulation to determine the amount of excess mechanical energy that can be absorbed as a function of step length and the corresponding eccentric joint work that can be generated in a single step we found that an increase in initial forward velocity and a greater forward shift of the com require a corresponding increase in the minimal step length needed for balance recovery furthermore the minimal step length is also a function of the muscle strength at the ankle the lower the muscle strength the greater the minimal step length required our theoretical framework reduces the complexity associated with previous studies relying on forward dynamics and iterative optimization processes this method may also be applied to study aspects of balance control such as the prevention of balance loss in the posterior or mediolateral direction

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

hamiltonian square roots of skew hamiltonian matrices revisited abstract recently h fassbender et al linear algebra appl 287 1999 125 proved the following theorem every real skew hamiltonian matrix w has a real hamiltonian square root h i e h 2 w we prove an analog of this theorem for complex matrices our approach may be of independent interest namely we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product

large vector spaces of block symmetric strong linearizations of matrix polynomials abstract given a matrix polynomial p i 0 k i a i of degree k where a i are n n matrices with entries in a field f the development of linearizations of p that preserve whatever structure p might posses has been a very active area of research in the last decade most of the structure preserving linearizations of p discovered so far are based on certain modifications of block symmetric linearizations the block symmetric linearizations of p available in the literature fall essentially into two classes linearizations based on the so called fiedler pencils with repetition which form a finite family and a vector space of dimension k of block symmetric pencils called dl p such that most of its pencils are linearizations one drawback of the pencils in dl p is that none of them is a linearization when p is singular in this paper we introduce new vector spaces of block symmetric pencils most of which are strong linearizations of p the dimensions of these spaces are o n 2 which for n k are much larger than the dimension of dl p when k is odd many of these vector spaces contain linearizations also when p is singular the coefficients of the block symmetric pencils in these new spaces can be easily constructed as k k block matrices whose n n blocks are of the form 0 i n a i or arbitrary n n matrices where is an arbitrary nonzero scalar

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

mobius transformations of matrix polynomials we discuss mobius transformations for general matrix polynomials over arbitrary fields analyzing their influence on regularity rank determinant constructs such as compound matrices and on structural features including sparsity and symmetry results on the preservation of spectral information contained in elementary divisors partial multiplicity sequences invariant pairs and minimal indices are presented the effect on canonical forms such as smith forms and local smith forms on relationships of strict equivalence and spectral equivalence and on the property of being a linearization or quadratification are investigated we show that many important transformations are special instances of mobius transformations and analyze a mobius connection between alternating and palindromic matrix polynomials finally the use of mobius transformations in solving polynomial inverse eigenproblems is illustrated

instability indices for matrix polynomials abstract there is a well established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are hermitian and skew hermitian this theory relates the number of negative directions for the matrix coefficients which are hermitian to the total number of unstable eigenvalues for the polynomial herein we extend the theory to even matrix polynomials of any finite degree in particular unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two we also consider hermitian matrix polynomials and derive an index which counts the number of eigenvalues with nonpositive imaginary part the results are refined if we consider the hermitian matrix polynomial to be a perturbation of a even polynomials however this refinement requires additional assumptions on the matrix coefficients

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

block kronecker ansatz spaces for matrix polynomials abstract in this paper we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangular matrix polynomial we provide a comprehensive characterization of the resulting vector spaces and show that almost every matrix pencil therein is a strong linearization regardless whether the matrix polynomial under consideration is regular or singular these novel ansatz spaces cover all block kronecker pencils as introduced in 6 as a subset and therefore contain all fiedler pencils modulo permutations the important case of square matrix polynomials is examined in greater depth we prove that the intersection of any number of block kronecker ansatz spaces is never empty and construct large subspaces of block symmetric matrix pencils among which still almost every pencil is a strong linearization moreover we show that the original ansatz spaces l 1 and l 2 may essentially be recovered from block kronecker ansatz spaces via pre and postmultiplication respectively of certain constant matrices

finding a positive definite linear combination of two hermitian matrices abstract let a and b be n by n hermitian matrices over the complex field a result of au yeung 1 and stewart 8 states that if x a ib x 0 for all nonzero n vectors x then there is a linear combination of a and b which is positive definite in this article we present an algorithm which finds such a linear combination in a finite number of steps we also discuss the implementation of the algorithm in case a and b are real symmetric sparse matrices

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

definite triples of hermitian matrices and matrix polynomials let a b and c be three n n nonzero hermitian matrices the triple a b c is called defnite if the convex hull of the joint numerical range f a b c x ax x bx x cx r3 x cn x x 1 does not contain 0 0 0 if the triple a b c is nondefinite then the numerical ranges of the matrix polynomials q a v3 b v2 c v1 v3 v2 v1 0 and l a 2 b ic 1 2 1 0 coincide with the whole complex plane providing no information as a consequence it is of particular interest to characterize a definite triple a b c and find the distance between 0 0 0 and the boundary f a b c the distance between a nondefinite triple a b c and the nearest definite triples with specified properties is also investigated moreover applications of definite triples on matrix polynomials of special interest are presented

automatic recovery of eigenvectors and minimal bases of matrix polynomials from generalized fiedler pencils with repetition abstract linearization is a widely used method for solving polynomial eigenproblems in which a matrix polynomial is transformed to a matrix pencil of larger size fiedler pencils generalized fiedler pencils fiedler pencils with repetition fprs and generalized fiedler pencils with repetition gfprs are well known classes of strong linearizations of matrix polynomials the class gfprs is also a rich source of structure preserving strong linearizations of structured matrix polynomials the recovery of eigenvectors minimal bases and minimal indices of matrix polynomials from those of the linearizations is an important task it is well known that eigenvectors and minimal bases of matrix polynomials can be easily recovered from those of the fiedler and generalized fiedler pencils by contrast barring a small subclass of fprs the recovery of eigenvectors and minimal bases of matrix polynomials from those of the fprs and gfprs is still an open problem the purpose of this paper is to fill this gap we consider the class of gfprs which subsumes the class of fprs and describe the recovery of eigenvectors minimal bases and minimal indices of matrix polynomials from those of the gfprs and show that the recovery is operation free

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

generic rank one perturbations of structured regular matrix pencils abstract classes of regular structured matrix pencils are examined with respect to their spectral behavior under a certain type of structure preserving rank 1 perturbations the observed effects are as follows on the one hand generically the largest jordan block at each eigenvalue gets destroyed or becomes of size one under a rank 1 perturbation depending on that eigenvalue occurring in the perturbating pencil or not on the other hand certain jordan blocks of t alternating matrix pencils occur in pairs so that in some cases the largest block cannot just be destroyed or shrunk to size one without violating the pairing thus the largest remaining jordan block will typically increase in size by one in these cases finally these results are shown to carry over to the classes of palindromic and symmetric matrix pencils

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

standard triples of structured matrix polynomials the notion of standard triples plays a central role in the theory of matrix polynomials we study such triples for matrix polynomials p lambda with structure mathcal s where mathcal s is the hermitian symmetric star even star odd star palindromic or star antipalindromic structure with star t we introduce the notion of mathcal s structured standard triple with the exception of t anti palindromic matrix polynomials of even degree with both 1 and 1 as eigenvalues we show that p lambda has structure mathcal s if and only if p lambda admits an mathcal s structured standard triple and moreover that every standard triple of a matrix polynomial with structure mathcal s is mathcal s structured we investigate the important special case of mathcal s structured jordan triples

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

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

linearizations of matrix polynomials in bernstein bases we discuss matrix polynomials expressed in a bernstein basis and the associated polynomial eigenvalue problems using mobius transformations of matrix polynomials large new families of strong linearizations are generated matrix polynomials that are structured with respect to a bernstein basis together with their associated spectral symmetries are also investigated the results in this paper apply equally well to scalar polynomials and include the development of new companion pencils for polynomials expressed in a bernstein basis

influence of simulated neuromuscular noise on movement variability and fall risk in a 3d dynamic walking model people at risk of falling exhibit increased gait variability which may predict future falls however the causal mechanisms underlying these correlations are not well known increased neuronal noise associated with aging likely leads to increased gait variability which could in turn lead to increased fall risk this paper presents a model of how changes in neuromuscular noise independently affect gait variability and probability of falling and aims to determine the extent to which changes in gait variability directly predict fall risk we used a dynamic walking model that incorporates a lateral step controller to maintain lateral stability noise was applied to this controller to approximate neuromuscular noise in humans noise amplitude was varied between low amplitudes that did not induce falls and high amplitudes for which the model always fell with increases in noise amplitude the model fell more often and after fewer steps gait variability increased with noise amplitude and predicted increased probability of falling importantly these relationships were not linear at either low gait variability or very high gait variability small increases in noise and variability affected probability of falling very little conversely at intermediate noise and or variability levels the same small increases resulted in large increases in probability of falling our results validate the idea that age related increases in neuromuscular noise likely play a direct contributing role in increasing fall risk however neuromuscular noise remains only one of many important factors that need to be considered these findings have important implications for fall prevention research and practice

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

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