on the segre weyr characteristics of right left regular matrix pencils the classical segre weyr characteristic theory of the standard eigenvalue eigenvector problem defined on a n n is extended to the case of right left regular pencils sf g f g m n the notions of toeplitz matrices and of the right left f g sequences j r f g j 1 f g respectively are introduced for right left regular pairs it is shown that j r f g j 1 f g are piecewise arithmetic progression sequences paps the singular points of those paps define the possible degrees of the elementary divisors e d at whereas the deviations or gaps from the arithmetic progression sequence define the multiplicity of the corresponding degree e d s the links between the algebraic notion of the segre characteristic and the geometric notion of the weyr characteristic are established for such families of pencils two different procedures for the computation of the segre characteristic are given the first is based on the analysis of singular points of paps and the second is based o

blocks of small defect abstract let g be a finite solvable group let p be a prime such that p 5 and o p g 1 and we denote g p p n then g contains a block of defect less than or equal to 3 n 5 let g be a finite solvable group and let p a be the largest power of p dividing 1 for an irreducible character of g we show that g f g p p 3 a for p 5

simple classical groups of lie type are determined by their character degrees let g be a finite group denote by irr g the set of all irreducible complex characters of g let cd g be the set of all irreducible complex character degrees of g forgetting multiplicities that is cd g 1 irr g and let x1 g be the set of all irreducible complex character degrees of g counting multiplicities let h be a finite nonabelian simple classical group in this paper we will show that if g is a finite group and x1 g x1 h then g is isomorphic to h in particular this implies that the nonabelian simple classical groups of lie type are uniquely determined by the structure of their complex group algebras

brauer characters and normal sylow p subgroups abstract in this paper we study some variations of the well known ito michler theorem for p brauer characters using various inequalities involving the p brauer character degrees of finite groups several new criteria for the existence of a normal sylow p subgroup of finite groups are obtained using the p parts and p parts of the p brauer character degrees

nearly defective complex eigenvalues in bessel matrices abstract it is known that an n n bessel matrix t n exhibits simple well separated complex eigenvalues which are getting more and more ill conditioned for increasing n the aim of this paper is to study the behavior of such eigenvalues which are simple in theory but in practice become defective using finite precision to this end we provide a sharp estimate of the distance of t n to the nearest defective matrix when n increases

characterization and construction of the nearest defective matrix via coalescence of pseudospectral components let a be a matrix with distinct eigenvalues and let w a be the distance from a to the set of defective matrices using either the 2norm or the frobenius norm define the pseudospectrum of a to be the set of points in the complex plane which are eigenvalues of matrices a e with kek and let c a be the supremum of all with the property that has n distinct components demmel and wilkinson independently observed in the 1980s that w a c a and equality was established for the 2 norm by alam and bora in 2005 we give new results on the geometry of the pseudospectrum near points

matrix polynomials with specified eigenvalues abstract this work concerns the distance in the 2 norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane initially we consider perturbations of the constant coefficient matrix only a singular value optimization characterization is derived for the associated distance we also consider the distance in the general setting when all of the coefficient matrices are perturbed in this general setting we obtain a lower bound in terms of another singular value optimization problem the singular value optimization problems derived facilitate the numerical computation of the distances

on condition numbers of polynomial eigenvalue problems in this paper we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients generalizing classical results of matrix perturbation theory we provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate we obtain that if a simple eigenvalue of a matrix polynomial is ill conditioned in some respects then it is close to be multiple and we construct an upper bound for this distance measured in the euclidean norm we also derive a new expression for the condition number of a simple eigenvalue which does not involve eigenvectors moreover an elsner like perturbation bound for matrix polynomials is presented

fixing multiple eigenvalues by a minimal perturbation abstract suppose we are given an n n matrix m and a set of values i i 1 m m n and we wish to find the smallest perturbation in the 2 norm i e spectral norm m such that m m has the given eigenvalues i some interesting results have been obtained for variants of this problem for fixing two distinct eigenvalues fixing one double eigenvalue and fixing a triple eigenvalue this paper provides a geometric motivation for these results and also motivates their generalization we also present numerical examples both successes and failures of fixing more than two eigenvalues by these generalizations

singular values and invariant factors of matrix sums and products abstract some remarks are made concerning the range of individual singular values of sums and products of two complex square matrices when these are allowed to vary in their orbits under unitary equivalence an analogous question is considered for invariant factors of products of matrices over a principal ideal domain

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

on the distance from a matrix polynomial to matrix polynomials with some prescribed eigenvalues abstract consider an n n matrix polynomial p and a set consisting of k n complex numbers recently kokabifar loghmani psarrakos and karbassi studied a weighted spectral norm distance from p to the n n matrix polynomials whose spectra contain the specified set under the assumption that all the entries of are distinct in this paper the case in which some or all of the desired eigenvalues can be multiple is discussed lower and upper bounds for the distance are computed and a perturbation of p associated to the upper bound is constructed a detailed numerical example illustrates the efficiency and validity of the proposed computational method

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

pseudospectra critical points and multiple eigenvalues of matrix polynomials abstract we develop a general framework for perturbation analysis of matrix polynomials more specifically we show that the normed linear space l m c n n of n by n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in l m c n n we present a family of natural norms on the space l m c n n and show that the norms on the spaces c m 1 and c n n play a crucial role in the perturbation analysis of matrix polynomials we define pseudospectra of matrix polynomials in the general framework of the normed space l m c n n and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases we analyze various properties of pseudospectra in the unified framework of the normed space l m c n n we analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial we show that common boundary points of components of pseudospectra of matrix polynomials are critical points as a consequence we show that a solution of wilkinson s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

a unified approach to computing the nearest complex polynomial with a given zero suppose we have a complex polynomial f z whose coefficients are inaccurate and a prescribed complex number such that f 0 we study the problem of computing a complex polynomial f z such that f 0 and the distance between f and f i e f f is minimal considering that previous works usually took the usual l p norm weighted l p norm and block wise norm as distance measures we first introduce a new defined synthetic norm that integrates all these norms then we propose a unified approach to study the proposed problem and succeed in giving explicit expressions of the nearest polynomial the effectiveness of our approach is illustrated by two examples one of which shows an extension of finding the nearest complex polynomial with a zero in a given domain finally as an application of the new defined norm we discuss a matrix valued optimization problem that is very common in machine learning this paper introduces a new synthetic block wise norm and proves its dual norm which is very important here this paper for the first time proposes a unified approach to compute the nearest complex polynomial with a given zero this paper discusses a matrix valued optimization problem that is very common in machine learning as an application of the introduced norm

parallel codes for computing the numerical rank abstract in this paper we present an experimental comparison of several numerical tools for computing the numerical rank of dense matrices the study includes the well known svd the urv decomposition and several rank revealing qr factorizations the qr factorization with column pivoting and two qr factorizations with restricted column pivoting two different parallel programming methodologies are analyzed in our paper first we present block partitioned algorithms for the urv decomposition and rank revealing qr factorizations which provide efficient implementations on shared memory environments furthermore we also present parallel distributed algorithms based on the message passing paradigm for computing rank revealing qr factorizations on multicomputers

perforated strict pseudospectra of demmel s matrices abstract regarding the pseudospectra of matrices it is easy to derive some conclusions from computer made graphics however those conclusions might be false without further analysis indeed it happened when discussing the presence of a local maximum at the origin for the least singular value of the characteristic matrix of the n n real toeplitz matrix whose first row and first column are 1 b b 2 b n 1 and 1 0 0 0 t respectively which was first taken into account by demmel in this paper we determine the value of a local maximum for the last singular value of the characteristic matrix of such toeplitz matrices in the cases n 3 and n 5 we also specify the point where this local maximum is attained at

finite group elements where no irreducible character vanishes in this paper we consider elements x of a finite group g with the property that x 0 for all irreducible characters of g if g is solvable and x has odd order we show that x must lie in the fitting subgroup f g

finite sequences of orthogonal polynomials connected by a jacobi matrix abstract if p0 pn is an orthogonal sequence with pj a monic polynomial of exact degree j all j then the sequence q0 qn of monic polynomials defined by the requirement that pj cjqn j 1pn 1 at the zeros of pn with cj suitable constants is also orthogonal with respect to a possibly differently weighted scalar product these two sequences can be shown to be the upper left respectively lower right principal minors of the matrix t j with j a suitable tridiagonal matrix the sequence q occurs in the characterization of the discrete least squares approximation to f from span p0 pj in terms of the n j 1 order divided differences of f p n 1 at the nodes i e the zeros of pn a complete characterization of the pair u v of zero sets of pj 1 and qn j is given and an application is made to the problem of recovering a jacobi matrix from such data

a formula for the 2 norm distance from a matrix to the set of matrices with multiple eigenvalues we prove that the 2 norm distance from an n times n matrix a to the matrices that have a multiple eigenvalue lambda is equal to where the singular values sigma k are ordered nonincreasingly therefore the 2 norm distance from a to the set of matrices with multiple eigenvalues is

on the length of the conjugacy classes of finite groups abstract let g be a finite group the question of how certain arithmetical conditions on the degrees of the irreducible characters of g influence the group structure was studied by several authors our purpose here is to impose analogous conditions on the lengths of the conjugacy classes of g and to describe the group structure under these conditions

on the construction of nearest defective matrices to a normal matrix given an n by n normal matrix a having n distinct eigenvalues we describe a simple procedure for constructing a defective matrix nearest to a our construction requires only an appropriate pair of eigenvalues of a and their corresponding eigenvectors

nearest matrix with two prescribed eigenvalues abstract given a square complex matrix a and two complex numbers z1 and z2 we find the distance from a to the set of matrices that have z1 z2 as some of their eigenvalues we use the distance between two matrices associated with the spectral norm

locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity the wilkinson distance of a matrix a is the two norm of the smallest perturbation e so that a e has a multiple eigenvalue malyshev derived a singular value optimization characterization for the wilkinson distance in this work we generalize the definition of the wilkinson distance as the two norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity we provide a singular value characterization for this generalized wilkinson distance then we outline a numerical technique to solve the derived singular value optimization problems in particular the numerical technique is applicable to malyshev s formula to compute the wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue

brief a fast algorithm for the computation of an upper bound on the µ norm a fast algorithm for the computation of the optimally frequency dependent scaled h norm of a finite dimensional lti system is presented it is well known that this quantity is an upper bound to the m norm furthermore it was recently shown to play a special role in the context of slowly time varying uncertainty numerical experimentation suggests that the algorithm generally converges quadratically

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

the distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue abstract for a matrix polynomial p and a given complex number we introduce a spectral norm distance from p to the matrix polynomials that have as an eigenvalue of geometric multiplicity at least and a distance from p to the matrix polynomials that have as a multiple eigenvalue then we compute the first distance and obtain bounds for the second one constructing associated perturbations of p

application of structured total least squares for system identification and model reduction the following identification problem is considered minimize the spl lscr sub 2 norm of the difference between a given time series and an approximating one under the constraint that the approximating time series is a trajectory of a linear time invariant system of a fixed complexity the complexity is measured by the input dimension and the maximum lag the question leads to a problem that is known as the global total least squares problem and alternatively can be viewed as maximum likelihood identification in the errors in variables setup multiple time series and latent variables can be considered in the same setting special cases of the problem are autonomous system identification approximate realization and finite time optimal spl lscr sub 2 model reduction the identification problem is related to the structured total least squares problem this paper presents an efficient software package that implements the theory the proposed method and software are tested on data sets from the database for the identification of systems daisy

finite nonsolvable groups in which only two nonlinear irreducible characters have equal degrees abstract y berkovich et al proc amer math soc 115 1992 955 959 classified finite groups in which the degrees of the nonlinear irreducible characters are distinct theorem 24 7 from y berkovich j algebra 184 1996 584 603 contains the classification of solvable groups in which only two nonlinear irreducible characters have equal degrees d 1 groups in this paper we obtain the classification of nonsolvable d 1 groups completing the classification of d 1 groups our proof depends on the classification of finite simple groups the results of the important paper illinois j math 33 no 1 1988 103 131 on rational simple groups play a key role as well

note on structured indefinite perturbations to hermitian matrices in a recent paper overton and van dooren have considered structured indefinite perturbations to a given hermitian matrix we extend their results to skew hermitian hamiltonian and skew hamiltonian matrices as an application we give a formula for computation of the smallest perturbation with a special structure which makes a given hamiltonian matrix own a purely imaginary eigenvalue

computational aspect to the nearest matrix with two prescribed eigenvalues abstract given a complex square matrix a and two complex numbers 1 and 2 we present a method to calculate the distance from a to the set of matrices x that have 1 and 2 as some of their eigenvalues we also find the nearest matrix x

on thompson s theorem let p be a prime divisor of the order of a finite group g thompson 1970 j algebra14 129 134 has proved the following remarkable result a finite group g is p nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p in fact in that case g is solvable in this note we prove that a group g having only one nonlinear irreducible character of p degree is a cyclic extension of thompson s group this result is a consequence of the following theorem a nonabelian simple group possesses two nonlinear irreducible characters 1 and 2 of distinct degrees such that p does not divide 1 1 2 1 here p is arbitrary but fixed our proof depends on the classification of finite simple groups some properties of solvable groups possessing exactly two nonlinear irreducible characters of p degree are proved some open questions are posed

level set methods for finding critical points of mountain pass type abstract computing mountain passes is a standard way of finding critical points we describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case we apply these techniques to describe a strategy for addressing the wilkinson problem of calculating the distance from a matrix to a closest matrix with repeated eigenvalues finally we relate critical points of mountain pass type to nonsmooth and metric critical point theory

on the orders of zeros of irreducible characters abstract let g be a finite group and p a prime number we say that an element g in g is a vanishing element of g if there exists an irreducible character of g such that g 0 the main result of this paper shows that if g does not have any vanishing element of p power order then g has a normal sylow p subgroup also we prove that this result is a generalization of some classical theorems in character theory of finite groups

nearest southeast submatrix that makes multiple a prescribed eigenvalue part 1 abstract given four complex matrices a b c and d where a c n n and d c m m and given a complex number z 0 what is the spectral norm distance from d to the set of matrices x c m m such that z 0 is a multiple eigenvalue of the matrix a b c x this problem is solved when z 0 is not an eigenvalue of a we also give a conjecture when z 0 is an eigenvalue of a

finite solvable groups with at most two nonlinear irreducible characters of each degree abstract the aim of this paper is to investigate the finite solvable groups with at most two nonlinear irreducible characters of each degree

structured low rank approximation this paper concerns the construction of a structured low rank matrix that is nearest to a given matrix the notion of structured low rank approximation arises in various applications ranging from signal enhancement to protein folding to computer algebra where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system the task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix this paper addresses some of the theoretical and numerical issues involved in the problem two procedures for constructing the nearest structured low rank matrix are proposed the procedures are flexible enough that they can be applied to any lower rank any linear structure and any matrix norm in the measurement of nearness the techniques can also be easily implemented by utilizing available optimization packages the special case of symmetric toeplitz structure using the frobenius matrix norm is used to exemplify the ideas throughout the discussion the concept rather than the implementation details is the main emphasis of the paper

groups whose prime graphs have no triangles abstract let g be a finite group and let cd g be the set of all complex irreducible character degrees of g let g be the set of all primes which divide some character degree of g the prime graph g attached to g is a graph whose vertex set is g and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of g in this paper we show that if g is a finite group whose prime graph g has no triangles then g has at most 5 vertices we also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups finally we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices

distance bounds for prescribed multiple eigenvalues of matrix polynomials abstract in this paper motivated by a problem posed by wilkinson we study the coefficient perturbations of a square matrix polynomial to a matrix polynomial that has a prescribed eigenvalue of specified algebraic multiplicity and index of annihilation for an n n matrix polynomial p and a given scalar c we introduce two weighted spectral norm distances e r and e r k from p to the n n matrix polynomials that have as an eigenvalue of algebraic multiplicity at least r and to those that have as an eigenvalue of algebraic multiplicity at least r and maximum jordan chain length exactly k respectively then we obtain a lower bound for e r k and derive an upper bound for e r by constructing an associated perturbation of p

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

nearest matrix with prescribed eigenvalues and its applications consider an n n matrix a and a set consisting of k n prescribed complex numbers lippert 2010 in a challenging article studied geometrically the spectral norm distance from a to the set of matrices whose spectra included specified set and constructed a perturbation matrix with minimum spectral norm such that a had in its spectrum this paper presents an easy practical computational method for constructing the optimal perturbation by improving and extending the methodology necessary definitions and lemmas of previous related works also some conceivable applications of this issue are provided

fixing two eigenvalues by a minimal perturbation abstract the optimal perturbation m to a matrix m such that m m has a given eigenvalue 0 is given by the eckart young theorem this perturbation is optimal in the sense that m 2 is minimal in this article we present a generalization finding m 2 optimal perturbations of m such that m m has two of given eigenvalues this result also generalizes recent work by malyshev

infinite elementary divisors of a matrix polynomial and implications a new characterization of infinite poles and zeros in terms of a natural extension of the idea of infinite elementary divisors of a matrix pencil to matrix polynomials is developed it is used to show that two previous descriptions of infinite frequency structure are equivalent subsequently a polynomial matrix transformation which simultaneously preserves the finite and infinite zero structure is presented

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

on p parts of character degrees and conjugacy class sizes of finite groups abstract let g be a finite group and irr g the set of irreducible complex characters of g let e p g be the largest integer such that p e p g divides 1 for some irr g we show that g f g p p k e p g for a constant k this settles a conjecture of a moreto 13 conjecture 4 we also study the related problems of the p parts of conjugacy class sizes of finite groups

spectral structures of the generalized companion form and applications in this note we investigate the relationship between the finite and infinite frequency structure of a regular polynomial matrix and that of a particular linearization called the generalized companion matrix a special resolvent decomposition of the regular polynomial matrix is proposed which is based on the weierstrass canonical form of this generalized companion matrix and the solution of a regular polynomial matrix description pmd is thus formulated from this resolvent decomposition both the initial conditions of the pseudostate and the input are considered

on neighbouring matrices with quadratic elementary divisors this paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrixa and relates them to those given by kakan it shows that all results of this kind may give severe overestimates of the distance to the nearest defective matrix and that matrices with elementary divisors of degrees higher than two are often at a much greater distance than those with quadratic divisors

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

the largest character degree and the sylow subgroups of finite groups let p be a sylow p subgroup and b g the largest irreducible character degree of a finite group g it is shown that p op g b g in general and p op g b g in case p is abelian

lie type like groups abstract a finite simple group of lie type in defining characteristic p has exactly two p blocks the principal block and a block of defect zero consisting of the steinberg character whose degree is the p part of the order of the group in this paper we characterize finite groups g which have exactly the principal p block and a p block of defect zero consisting of an irreducible character of degree g p

numerical computation of an analytic singular value decomposition of a matrix valued function this paper extends the singular value decomposition to a path of matricese t an analytic singular value decomposition of a path of matricese t is an analytic path of factorizationse t x t s t y t t wherex t andy t are orthogonal ands t is diagonal to maintain differentiability the diagonal entries ofs t are allowed to be either positive or negative and to appear in any order this paper investigates existence and uniqueness of analytic svd s and develops an algorithm for computing them we show that a real analytic pathe t always admits a real analytic svd a full rank smooth pathe t with distinct singular values admits a smooth svd we derive a differential equation for the left factor develop euler like and extrapolated euler like numerical methods for approximating an analytic svd and prove that the euler like method converges

on pseudospectra of matrix polynomials and their boundaries in the first part of this paper sections 2 4 the main concern is with the boundary of the pseudospectrum of a matrix polynomial and particularly with smoothness properties of the boundary in the second part sections 5 6 results are obtained concerning the number of connected components of pseudospectra as well as results concerning matrix polynomials with multiple eigenvalues or the proximity to such polynomials

structured pseudospectra and structured sensitivity of eigenvalues pseudospectra and structured pseudospectra have been investigated widely in this paper for the matrix spectral norm we give a new equivalent definition of structured pseudospectra of a matrix triplet a b c via the so called restricted singular value decomposition rsvd according to this definition we provide a solution of structured wilkinson s problem which concerns structured sensitivity of eigenvalues of square matrices several examples are given to illustrate our results also we extend the new definition of structured pseudospectra to polynomial eigenvalue problems

on the distance to singular descriptor dynamical systems with impulsive initial conditions in this paper we study the problem of computing the distance between a given singular descriptor system e a and a nearest descriptor system that has impulsive initial conditions the link between existence of impulsive initial conditions and zeros at infinity for the associated matrix pencil se a is well known much of the literature focusses on the case when only one of e and a is perturbed we give a closed form expression of the distance to a nearest descriptor system having impulsive solutions via rank 1 perturbations when both e and a are perturbed next for the case of perturbations without rank restrictions we propose and evaluate the bounds for the distance in the context of structured perturbations we formulate and obtain an explicit expression for the distance when e and a are hermitian and are perturbed by hermitian matrices for a suitable class of systems we also show that upper and lower bounds are within a factor of sqrt 2 we finally construct examples and compare the bounds obtained from our results with those from the literature as well as with computed values of the distance obtained via three numerical optimization techniques such as the structured low rank approximation the broyden fletcher goldfarb shanno algorithm and direct optimization tools like globalsearch

on the computation of the jordan canonical form of regular matrix polynomials in this paper an algorithm for the computation of the jordan canonical form of regular matrix polynomials is proposed the new method contains rank conditions of suitably defined block toeplitz matrices and it does not require the computation of the jordan chains or the smith form the segre and weyr characteristics are also considered

nearest matrix with a prescribed eigenvalue of bounded multiplicities abstract let a c n n and z 0 c be given we compute the distance with respect to the spectral norm from a to the set of matrices that have z 0 as an eigenvalue of algebraic multiplicity at least 3 and geometric multiplicity at least 2 with an additional hypothesis on the maximum of a function furthermore we determine a nearest matrix b to a in this set

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

the interplay of ranks of submatrices a banded invertible matrix t has a remarkable inverse all upper and lower submatrices of t 1 have low rank depending on the bandwidth in t the exact rank condition is known and it allows fast multiplication by full matrices that arise in the boundary element method we look for the right proof of this property of t 1 ultimately it reduces to a fact that deserves to be better known complementary submatrices of any t and t 1 have the same nullity the last figure in the paper when t is tridiagonal shows that on and above the diagonal of t 1 all rows are proportional

on sensitivity of eigenvalues and eigendecompositions of matrices abstract the main purpose of this paper is to provide solutions of two problems on sensitivity of eigenvalues and eigendecompositions of matrices the first problem is due to wilkinson and it concerns finding the distance from an n n matrix with n distinct eigenvalues to the set of matrices having multiple eigenvalues we also describe how to construct a nearest matrix having a multiple eigenvalue the second problem concerns providing a characterization of the stability of eigendecompositions of matrices and is due to demmel

on perturbations of matrix pencils with real spectra ii a well known result on spectral variation of a hermitian matrix due to mirsky is the following let a and a be two n x n hermitian matrices and let 1 n and 1 n be their eigenvalues arranged in ascending order then diag 1 1 n n a a for any unitarily invariant norm in this paper we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra the much studied case of definite pencils is included in this

perturbation and robust stability of autonomous singular linear matrix difference equations in the perturbation theory of linear matrix difference equations it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear in this paper the perturbation theory of complex weierstrass canonical form for regular matrix pencils is investigated by using matrix pencil theory and the weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil moreover we study robust stability of a class of linear matrix difference equations of first and higher order whose coefficients are square constant matrices