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

uncontrollability for linear autonomous multi input dynamical systems depending on parameters linear multi input dynamical systems smoothly depending on parameters are considered a set of parameter values corresponding to uncontrollable systems an uncontrollability set is studied the typical generic structure of the uncontrollability set is described a constructive method of perturbation analysis of the uncontrollability set is developed formulae of first order approximations for the uncontrollability set and generalized eigenvalues uncontrollable modes are derived and used for numerical construction of the uncontrollability set the method is based on the versal deformation theory for matrix pairs under feedback equivalence as an example the uncontrollability set is found for a three parameter two degree of freedom mechanical system

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

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

perturbation of linear control systems abstract we study the variation of the controllability indices and the jordan structure of a pair of matrices a b under small perturbations

generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree abstract the set pol d r m n of m n complex matrix polynomials of grade d and normal rank at most r in a complex d 1 m n dimensional space is studied for r 1 min m n 1 we show that pol d r m n is the union of the closures of the r d 1 sets of matrix polynomials with rank r degree exactly d and explicitly described complete eigenstructures in addition for the full rank rectangular polynomials i e r min m n and m n we show that pol d r m n coincides with the closure of a single set of the polynomials with rank r degree exactly d and the described complete eigenstructure these complete eigenstructures correspond to generic m n matrix polynomials of grade d and rank at most r

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

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

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

change of the congruence canonical form of 2 by 2 and 3 by 3 matrices under perturbations and bundles of matrices under congruence we construct the hasse diagrams g2 and g3 for the closure ordering on the sets of congruence classes of 2 2 and 3 3 complex matrices in other words we construct two directed graphs whose vert

stratification of linear systems bifurcation diagrams for families of linear systems abstract a dynamical system can be represented by x ax bu y cx where a is a square matrix and b c are rectangular matrices the question of uncertain parameters e in the entries of the matrices a b c is particularly important when using the kronecker form of the triple of matrices a b c the eigenstructure may depend discontinuously on the parameters when the matrices a e b e c e depend smoothly on those parameters it is of great interest to know which different structures can arise from small perturbations of a dynamical system and discuss the generic behaviour of smooth few parameter families of linear systems a fundamental way of dealing with these problems is in a first step to stratify the space of triples of matrices defining the systems here an important role is played by the miniversal deformations a second step is to induce a partition in the space of parameters parametrizing the family of linear systems we need to consider transversal families in order to ensure that the induced partition called the bifurcation diagram is also a stratification in this case the induced partition is called a bifurcation diagram

a complete characterization of orbit closures of controllable singular systems under restricted system equivalence in this paper properties of the orbit space of controllable generalized state space systems modulo restricted system equivalence are derived in particular it is shown that this space is a smooth quasiprojective variety of dimension nm then the possible degenerations of controllable systems under transformations of restricted system equivalence are characterized and it is proved that every noncontrollable system can be approximated by a family of controllable systems that belong to a single equivalence class in the single input case this class is uniquely determined whereas in the multivariable case a noncontrollable system may lie in the boundary of finitely or even infinitely many equivalence classes

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

trimmed linearizations for structured matrix polynomials we discuss the eigenvalue problem for general and structured matrix polynomials which may be singular and may have eigenvalues at infinity we derive condensed forms that allow partial deflation of the infinite eigenvalue and singular structure of the matrix polynomial the remaining reduced order staircase form leads to new types of linearizations which determine the finite eigenvalues and corresponding eigenvectors the new linearizations also simplify the construction of structure preserving linearizations

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

on a homeomorphism between orbit spaces of linear systems and matrix polynomials abstract the orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact open topology is considered in the latter as a consequence the variation of the finite and left wiener hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described

local perturbations of generalized systems under feedback and derivative feedback following v i arnolds techniques we construct a local canonical form called miniversal deformation for a family of differentiable regularizable linear dynamical systems ex t ax t bu t under feedback and derivative feedback equivalence we show some applications to the analysis of perturbations of a given regularizable system

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

on perturbations and the equivalence orbit of a matrix pencil abstract two theorems on the canonical kronecker form of a perturbed matrix pencil and the characterization of the closure of the set of all matrix pencils with a fixed kronecker canonical form are given

the dimension of matrices matrix pencils with given jordan kronecker canonical forms abstract the set of n by n matrices with a given jordan canonical form defines a subset of matrices in complex n 2 dimensional space we analyze one classical approach and one new approach to count the dimension of this set the new approach is based upon and meant to give insight into the staircase algorithm for the computation of the jordan canonical form as well as the occasional failures of this algorithm we extend both techniques to count the dimension of the more complicated set defined by the kronecker canonical form of an arbitrary rectangular matrix pencil a b

robustness and perturbations of minimal bases ii the case with given row degrees abstract this paper studies generic and perturbation properties inside the linear space of m m n polynomial matrices whose rows have degrees bounded by a given list d 1 d m of natural numbers which in the particular case d 1 d m d is just the set of m m n polynomial matrices with degree at most d thus the results in this paper extend to a much more general setting the results recently obtained in 29 only for polynomial matrices with degree at most d surprisingly most of the properties proved in 29 as well as their proofs remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well known sylvester matrices of polynomial matrices are replaced by the new trimmed sylvester matrices introduced in this paper the following results are presented among many others in this work 1 generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to d 1 d m and with right minimal indices differing at most by one and having a sum equal to i 1 m d i and 2 under perturbations these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly

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

the algebraic structure of pencils and block toeplitz matrices abstract we restate several results on the kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation and state some new results on the kronecker structure after a new row is appended or a column deleted the results are simple consequences of a theory of majorization for semi infinite integer sequences the bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time invariant dynamical system we illustrate the known relations between the kronecker indices the segre characteristics and the nullities of certain block toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row

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

the eigenstructure of an arbitrary polynomial matrix computational aspects we give a new numerical method to compute the eigenstructure i e the zero structure the polar structure and the left and right null space structure of a polynomial matrix p these structural elements are of fundamental importance in several systems and control problems involving polynomial matrices the approach is more general than previous numerical methods because it can be applied to an arbitrary m n polynomial matrix p with normal rank r smaller than m and or n the algorithm is then shown to compute the structure of the left and right null spaces of p as well the speed and accuracy of this new approach are also discussed

automatic grid generation and flow solution for complex geometries the development of a new method for the automatic generation of computational grids around complex aerodynamic configurations is described the grid generation procedure uses a cartesian cell embedded unstructured approach that enables the efficient placement of computational cells to resolve important flow regions close to the surface the main difference between the present method and most existing grid generation techniques is the avoidance of the need to explicitly generate a surface grid this new approach is easily automated and is capable of producing grids around arbitrarily complex geometries the capabilities of the method are demonstrated with a selection of inviscid transonic flows solutions