arveson s criterion for unitary similarity abstract this paper is an exposition of w b arveson s complete invariant for the unitary similarity of complex irreducible matrices

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

unitary similarity to a normal matrix abstract we give several criteria of unitary similarity of a normal matrix a and any matrix b in terms of the frobenius and spectral norms characteristic polynomials and traces of matrices

overlapping schwarz preconditioners for indefinite time harmonic maxwell equations time harmonic maxwell equations in lossless media lead to a second order differential equation for the electric field involving a differential operator that is neither elliptic nor definite a galerkin method using nedelec spaces can be employed to get approximate solutions numerically the problem of preconditioning the indefinite matrix arising from this method is discussed here specifically two overlapping schwarz methods will be shown to yield uniform preconditioners

the intersection of the similarity and conjunctivity equivalence classes abstract let a be an n n complex matrix let sim a denote the similarity equivalence class of a let conj a denote the conjunctivity equivalence class of a and let u a denote the unitary similarity equivalence class of a define cs a sim a conj a we seek to classify the matrices that have cs a u a and show by an example that this is not true in general but we show that it is true when a is hermitian or is a scalar multiple of a hermitian for n 3 we reduce the general n n case to the case when a is non singular and not a multiple of a hermitian matrix we completely classify the 2 2 case and find that cs a u a when a is non singular or normal or both

analysis of a finite element method for maxwell s equations the use of finite elements to discretize the time dependent maxwell equations on a bounded domain in three dimensional space is analyzed energy norm error estimates are provided when general finite element methods are used to discretize the equations in space in addition it is shown that if some curl conforming elements due to nedelec are used error estimates may also be proved in the l 2 norm

calculation of all stabilizing fractional order pd controllers for integrating time delay systems in this paper a simple and effective stabilization method for integrating time delay systems using fractional order pd controllers c s k p k ds m is proposed the presented method is based on finding the stability regions according to the fractional orders of the derivative element in the range of 0 2 these regions are computed by using three stability boundaries real root boundary rrb complex root boundary crb and infinite root boundary irb the method gives the explicit formulae corresponding to these boundaries in terms of fractional order pd controller pd m controller parameters thus the complete set of stabilizing controllers for an arbitrary integrating time delay system can be obtained in order to demonstrate the effectiveness in solution accuracy and the simplicity of this method two simulation studies are given the simulation results indicate that the pd m controllers can provide larger stability regions than the integer order pd controllers

nonoverlapping domain decomposition methods with a simple coarse space for elliptic problems we propose a substructuring preconditioner for solving three dimensional elliptic equations with strongly discontinuous coefficients the new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by bramble pasiack and schatz 1989 but with much simpler coarse solvers though the condition number of the preconditioned system may not have a good bound we are able to show that the convergence rate of the pcg method with such substructuring preconditioner is nearly optimal and also robust with respect to the possibly large jumps of the coefficient in the elliptic equation

a mortar edge element method with nearly optimal convergence for three dimensional maxwell s equations in this paper we are concerned with mortar edge element methods for solving three dimensional maxwell s equations a new type of lagrange multiplier space is introduced to impose the weak continuity of the tangential components of the edge element solutions across the interfaces between neighboring subdomains the mortar edge element method is shown to have nearly optimal convergence under some natural regularity assumptions when nested triangulations are assumed on the interfaces a generalized edge element interpolation is introduced which plays a crucial role in establishing the nearly optimal convergence the theoretically predicted convergence is confirmed by numerical experiments

simultaneous reduction of sets of matrices under similarity abstract let s be a set of n n matrices over a field f and a the algebra generated by s over f the problem of deciding whether the elements of s can be simultaneously reduced to block triangular form with the diagonal blocks of some specified size is considered and an account is given of various methods used to attack the problem most of the techniques use representation theory to obtain information on a the problems of simultaneous triangularization existence of common eigenvectors etc are also considered the aim of the paper is to survey the methods used to attack these problems and to give some typical results the paper does not contain many new results

a conjectured property of hermitian pencils abstract if a b is any pair of hermitian matrices the power of dividing det i xa yb will be given by the number of basic singular summands in the pair contrary to conjecture this power can be greater than one even when the pair is unitarily irreducible

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

least squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups let s a denote the orbit of a complex or real matrix a under a certain equivalence relation such as unitary similarity unitary equivalence unitary congruences etc efficient gradient flow algorithms are constructed to determine the best approximation of a given matrix a0 by the sum of matrices in s a1 s an in the sense of finding the euclidean least squares distance

technical communique fractional order proportional derivative controller for a class of fractional order systems recently fractional order systems fos have attracted more and more attention in various fields but the control design techniques available for the fos suffer from the lack of direct systematic approaches in this paper we focus on a given type of simple model of fos a fractional order proportional derivative fo pd controller is proposed for this class of fos and a practical and systematic tuning procedure has been developed for the proposed fo pd controller synthesis the fairness issue in comparing with other controllers such as the traditional integer order pid io pid controller and the fractional order proportional derivative fo pd controller has been addressed under the same number of design parameters and the same specifications fair comparisons of the three controllers i e io pid fo pd and fo pd via the simulation tests illustrate that the io pid controller designed may not always be stabilizing to achieve flat phase specification while both fo pd and fo pd controllers designed are always stabilizing furthermore the proposed fo pd controller outperforms fo pd controller for the class of fractional order systems

operators on positive semidefinite inner product spaces abstract let u be a semiunitary space i e a complex vector space with scalar product given by a positive semidefinite hermitian form if a linear operator a u u is bounded i e a u c u for some c r and all u u then the subspace u 0 u u u u 0 is invariant and so a defines the linear operators a 0 u 0 u 0 and a 1 u u 0 u u 0 let a be an indecomposable bounded operator on u such that 0 u 0 u let be an eigenvalue of a 0 we prove that the algebraic multiplicity of in a 1 is not less than the geometric multiplicity of in a 0 and the geometric multiplicity of in a 1 is not less than the number of jordan blocks j t of each fixed size t t in the jordan canonical form of a 0 we give canonical forms of selfadjoint and isometric operators on u and of hermitian forms on u for an arbitrary system of semiunitary spaces and linear mappings on between them we give an algorithm that reduces their matrices to canonical form its special cases are belitskii s and littlewood s algorithms for systems of linear operators on vector spaces and unitary spaces respectively

a bddc algorithm with deluxe scaling for three dimensional h curl problems in our paper we present and analyze a bddc algorithm for a class of elliptic problems in the three dimensional h curl space compared with existing results our condition number estimate requires fewer assumptions and also involves two fewer powers of log h h making it consistent with optimal estimates for other elliptic problems here h his the maximum of hi hi over all subdomains where hi and hi are the diameter and the smallest element diameter for the subdomain i the analysis makes use of two recent developments the first is our new approach to averaging across the subdomain interfaces while the second is a new technical tool that allows arguments involving trace classes to be avoided furthermore numerical examples are presented to confirm the theory and demonstrate the importance of the new averaging approach in certain cases

isometries of the toeplitz matrix algebra abstract we study the structure of isometries defined on the algebra a of upper triangular toeplitz matrices our first result is that a continuous multiplicative norm preserving map a m n c must be of the form either a u a u or a u a u where a is the complex conjugation and u is a unitary matrix in our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry a m n c is of the form a u a v where u and v are two unitary matrices this implies in particular that every such an isometry is a complete isometry and that a unital linear isometry a m n c is necessarily an algebra homomorphism

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

on a finite element method for solving the three dimensional maxwell equations abstract the aim of this paper is to present a method for solving the time domain three dimensional maxwell equations which can be coupled with a particle solver for this purpose maxwell s equations are reformulated as a constrained wave equation system with lagrange multipliers associated to the conditions 0 and e e 0 we approximate both the fields and the lagrange multipliers with a finite element method using a taylor hood element

a robust preconditioned minres solver for distributed time periodic eddy current optimal control problems this work is devoted to the construction and analysis of robust solvers for some distributed optimal control problems for time periodic eddy current problems we apply the multiharmonic approach to the optimality system and construct a new preconditioned minres solver for the system of frequency domain equations we show that this solver is robust with respect to the space discretization and time discretization parameters as well as all involved bad parameters like the conductivity the reluctivity and the regularization and cost parameters

an iterative substructuring method for raviart thomas vector fields in three dimensions the iterative substructuring methods also known as schur complement methods form one of two important families of domain decomposition algorithms they are based on a partitioning of a given region on which the partial differential equation is defined into nonoverlapping substructures the preconditioners of these conjugate gradient methods are then given in terms of local problems defined on individual substructures and pairs of substructures and in addition a global problem of low dimension an iterative method of this kind is introduced for the lowest order raviart thomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure the theoretical bounds are confirmed by a series of numerical experiments

pairs of matrices with quadratic minimal polynomials abstract let a b be n n matrices over a field f and suppose a b have quadratic minimal polynomials then the algebra generated by a b has dimension at most 2n 1 if n is odd and 2n if n is even these bounds are exact

specht s criterion for systems of linear mappings abstract w specht 1940 proved that two n n complex matrices a and b are unitarily similar if and only if trace w a a trace w b b for every word w x y in two noncommuting variables we extend his criterion and its generalizations by n a wiegmann 1961 and n jing 2015 to an arbitrary system a consisting of complex or real inner product spaces and linear mappings among them we represent such a system by the directed graph q a whose vertices are inner product spaces and arrows are linear mappings denote by q a the directed graph obtained by enlarging to q a the adjoint linear mappings we prove that a system a is transformed by isometries of its spaces to a system b if and only if the traces of all closed directed walks in q a and q b coincide

a counterexample to kippenhahn s conjecture on hermitian pencils abstract a counterexample is constructed to a conjecture of kippenhahn math nachr 6 193 288 1951 52 a pair of hermitian 8 8 matrices h k is found such that 1 h k generate m 8 c and 2 the minimal polynomial of the pencil xh yk has degree 4 recent work of h shapiro e g linear algebra appl 43 201 221 1982 has established the conjecture for n n matrices n 5

stabilisability analysis of high order unstable processes by fractional order controllers in this article the sufficient stabilisability criteria for unstable time delay processes by fractional order controllers are investigated the process is a high order system with a single unstable pole which also has a stable zero the adopted approach to derive the sufficient conditions for stability is based on the well known nyquist stability criterion stabilisability is studied by applying fractional order proportional integral and proportional derivative controllers and the results are given in terms of the maximum allowable value of time delay additionally limitations on the parameters of the controllers which must be taken into account in the controller design are proposed

a structure theorem for some matrix algebras abstract let f be a field and m n f the algebra of n n matrices over f it is in general a very difficult and tedious problem to determine the structure of the subalgebra x of m n f generated by a given subset s of m n f we show that if x contains a matrix a with n distinct eigenvalues in f then x can be determined up to similarity very quickly by a graph theoretic method as a consequence we show that any such x can be generated by a pair of elements one of which can be taken to be a and the other a 0 1 matrix as an application we obtain a specht type similarity theorem

a nonoverlapping domain decomposition method for maxwell s equations in three dimensions in this paper we propose a nonoverlapping domain decomposition method for solving the three dimensional maxwell equations based on the edge element discretization for the schur complement system on the interface we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping subdomains it is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients

a criterion for unitary similarity of upper triangular matrices in general position abstract each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order let a a ij and b b ij be upper triangular n n matrices that are not similar to direct sums of square matrices of smaller sizes or are in general position and have the same main diagonal we prove that a and b are unitarily similar if and only if h a k h b k for all h c x and k 1 n where a k a ij i j 1 k and b k b ij i j 1 k are the leading principal k k submatrices of a and b and is the frobenius norm

substructuring preconditioners for saddle point problems arising from maxwell s equations in three dimensions this paper is concerned with the saddle point problems arising from edge element discretizations of maxwell s equations in a general three dimensional nonconvex polyhedral domain a new augmented technique is first introduced to transform the problems into equivalent augmented saddle point systems so that they can be solved by some existing preconditioned iterative methods then some substructuring preconditioners are proposed with very simple coarse solvers for the augmented saddle point systems with the preconditioners the condition numbers of the preconditioned systems are nearly optimal namely they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size

on the congruence transformation of a pencil of real symmetric matrices to a pencil with identical characteristic polynomial abstract simultaneous nonorthogonal congruence transformations for pairs a b of 2 2 real symmetric matrices which leave the characteristic polynomial of the pencil x 1 a x 2 b invariant are studied

a generalization of the motzkin taussky theorem abstract in this paper we generalize the motzkin taussky theorem to matrices with polynomial entries

the constrained bilinear form and the c numerical range abstract let v be an n dimentional unitary space with inner product and s the set x v x x 1 for any a hom v v and q c with q 1 we define w a q ax y x y s x y q if q 1 then w a q is just the classical numerical range ax x x s the convexity of which is well known another generalization of the numerical range is the c numerical range which is defined to be the set w c a tr cu au u unitary where c hom v v in this note we prove that w a q is always convex and that wc a is convex for all a if rank c 1 or n 2

simultaneous state estimation and parameter identification in linear fractional order systems using coloured measurement noise abstractin the present paper the identification and estimation problem of a single input single output siso fractional order state space system will be addressed a siso state space model is considered in which parameters and also state variables should be estimated the canonical fractional order state space system will be transformed into a regression equation by using a linear transformation and a shift operator that are appropriate for identification the identification method provided in this paper is based on a recursive identification algorithm that has the capability of identifying the parameters of fractional order state space system recursively another subject that will be addressed in this paper is a novel fractional order kalman filter suitable for the systems with coloured measurement noise the promising performance of the proposed methods is verified using two stable fractional order systems

auxiliary space preconditioning in h 0 curl ω we adapt the principle of auxiliary space preconditioning as presented in j xu the auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids computing 56 1996 pp 215 235 to h curl elliptic variational problems discretized by means of edge elements the focus is on theoretical analysis within the abstract framework of subspace correction employing a helmholtz type splitting of edge element vector fields we can establish asymptotic h uniform optimality of the preconditioner defined by our auxiliary space method

unitarily achievable zero patterns and traces of words in a and a abstract our primary objective is to identify a natural and substantial problem about unitary similarity on arbitrary complex matrices which 0 patterns may be achieved for any given n by n complex matrix via some unitary similarity of it to this end certain restrictions on achievable 0 patterns are mentioned both positional and more important on the maximum number of achievable 0 s prior results fitting this general question are mentioned as well as the first unresolved pattern for 3 by 3 matrices in the process a recent question is answered a closely related additional objective is to mention the best known bound for the maximum length of words necessary for the application of specht s theorem about which pairs of complex matrices are unitarily similar which seems not widely known to matrix theorists in the process we mention the number of words necessary for small size matrices

on the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence abstract let a 1 a 2 be given n by n hermitian or symmetric matrices and consider the simultaneous transformations a i sa i s if a i is hermitian or a i sa i s t if a i is symmetric we give necessary and sufficient conditions for the existence of a unitary s which reduces both a 1 and a 2 to diagonal form in this way when at least one of a 1 or a 2 is nonsingular we give necessary and sufficient conditions for a reduction of this sort with a nonsingular s these results are a generalization of the classical theorem from mechanics that a positive definite matrix and a hermitian matrix can always be diagonalized simultaneously by a nonsingular congruence

fractional delayed damped mathieu equation this paper investigates the dynamical behaviour of the fractional delayed damped mathieu equation this system includes three different phenomena fractional order time delay parametric resonance the method of harmonic balance is employed to achieve approximate expressions for the transition curves in the parameter plane the n 0 and n 1 transition curves both lower and higher order approximations are obtained the dependencies of these curves on the system parameters and fractional orders are determined previous results for the transition curves reported for the damped mathieu equation delayed second order oscillator and fractional mathieu equation are confirmed as special cases of the results for the current system

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

overlapping schwarz methods for maxwell s equations in three dimensions two level overlapping schwarz methods are considered for finite element problems of 3d maxwell s equations nedelec elements built on tetrahedra and hexahedra are considered once the relative overlap is fixed the condition number of the additive schwarz method is bounded independently of the diameter of the triangulation and the number of subregions a similar result is obtained for a multiplicative method these bounds are obtained for quasi uniform triangulations in addition for the dirichlet problem the convexity of the domain has to be assumed our work generalizes well known results for conforming finite elements for second order elliptic scalar equations

matrices with some extremal properties abstract for a complex matrix a there are many inequalities related to its eigenvalues diagonal elements and three types of singular values we investigate the conditions on the matrices for which those inequalities become equalities it is shown that in many cases the conditions are both necessary and sufficient for the matrices to be unitarily similar to matrices of lower orders moreover we get alternative proofs of various characterization theorems of different classes of matrices

a domain decomposition algorithm for elliptic problems in three dimensions most domain decomposition algorithms have been developed for problems in two dimensions one reason for this is the difficulty in devising a satisfactory easy to implement robust method of providing global communication of information for problems in three dimensions several methods that work well in two dimension do not perform satisfactorily in three dimensions a new iterative substructuring algorithm for three dimensions is proposed it is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain the condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains the new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions

a discrete weighted helmholtz decomposition and its application we propose a discrete weighted helmholtz decomposition for edge element functions the decomposition is orthogonal in a weighted l 2 inner product and stable uniformly with respect to the jumps in the discontinuous weight function as an application the new helmholtz decomposition is applied to demonstrate the quasi optimality of a preconditioned edge element system for solving a saddle point maxwell system in non homogeneous media by a non overlapping domain decomposition preconditioner i e the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain and more importantly it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media numerical experiments are presented to validate the effectiveness of the non overlapping domain decomposition preconditioner

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

convergence analysis of a covolume scheme for maxwell s equations in three dimensions this paper contains error estimates for covolume discretizations of maxwell s equations in three space dimensions several estimates are proved first an estimate for a semi discrete scheme is given second the estimate is extended to cover the classical interlaced time marching technique third some of our unstructured mesh results are specialized to rectangular meshes both uniform and nonuniform by means of some additional analysis it is shown that the spatial convergence rate is one order higher than for the unstructured case

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

an e based splitting method with a non matching grid for eddy current equations with discontinuous coefficients a new decoupled finite element method based on solving a vector and a scalar from the splitting of the electric field is presented to approximate time dependent eddy current equations with discontinuous coefficients in a three dimensional polyhedral domain by introducing a lagrangian multiplier this method realizes an optimal control for the interface problem a non matching finite element grid on the interface is considered and an optimal energy norm error estimate in finite time is obtained

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

on the gau wu number for some classes of matrices abstract for a given n n matrix a let k a stand for the maximal number of orthonormal vectors x j such that the scalar products a x j x j lie on the boundary of the numerical range w a this number was recently introduced by gau and wu and we therefore call it the gau wu number of the matrix a we compute k a for two classes of n n matrices a a simple and explicit expression for k a for tridiagonal toeplitz matrices a is derived furthermore we prove that k a 2 for every pure almost normal matrix a note that for every matrix a we have k a 2 and for normal matrices a we have k a n so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the gau wu number finally matrices with maximal gau wu number k a n are considered

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

an optimal domain decomposition preconditioner for low frequency time harmonic maxwell equations the time harmonic maxwell equations are considered in the low frequency case a finite element domain decomposition approach is proposed for the numerical approximation of the exact solution this leads to an iteration by subdomain procedure which is proven to converge the rate of convergence turns out to be independent of the mesh size showing that the preconditioner implicitly defined by the iterative procedure is optimal for obtaining this convergence result it has been necessary to prove a regularity theorem for dirichlet and neumann harmonic fields

unitary and euclidean representations of a quiver abstract a unitary euclidean representation of a quiver is given by assigning to each vertex a unitary euclidean vector space and to each arrow a linear mapping of the corresponding vector spaces we recall an algorithm for reducing the matrices of a unitary representation to canonical form give a certain description of the representations of canonical form and reduce the problem of classifying euclidean representations to the problem of classifying unitary representations we also describe the set of dimensions of all indecomposable unitary euclidean representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

an iterative substructuring method for maxwell s equations in two dimensions iterative substructuring methods also known as schur complement methods form an important family of domain decomposition algorithms they are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner for conforming finite element approximations of h 1 it is known that the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure in this paper the same result is established for similar iterative methods for low order n e d e lec finite elements which approximate hcurl in two dimensions results of numerical experiments are also provided

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

on a conjecture of kippenhahn about the characteristic polynomial of a pencil generated by two hermitian matrices i abstract let a be an n n matrix write a h ik where i 2 1 and h and k are hermitian let f x y z det zi xh yk we first show that a pair of matrices over an algebraically closed field which satisfy quadratic polynomials can be put into block upper triangular form with diagonal blocks of size 1 1 or 2 2 via a simultaneous similarity this is used to prove that if f x y z g x y z n 2 where g has degree 2 then for some unitary matrix u the matrix u au is the direct sum of n 2 copies of a 2 2 matrix a 1 where a 1 is determined up to unitary similarity by the polynomial g x y z we use the connection between f x y z and the numerical range of a to investigate the case where f x y z has the form z ax y r g x y z s where g x y z is irreducible of degree 2

a complete unitary similarity invariant for unicellular matrices abstract we present necessary and sufficient conditions for an n n complex matrix b to be unitarily similar to a fixed unicellular i e indecomposable by similarity n n complex matrix a

a feti domain decomposition method for edge element approximations in two dimensions with discontinuous coefficients a class of finite element tearing and interconnecting feti methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed first an abstract framework is presented for the analysis of a class of feti methods where a natural coarse problem associated with the substructures is lacking then a family of feti methods for edge element approximations is proposed it is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures the estimate is also independent of the jumps of both of the coefficients of the original problem numerical results validating our theoretical bounds are given the method and its analysis can be easily generalized to raviart thomas element approximations in two and three dimensions

hermitian pencils with a cubic minimal polynomial let a be an n n complex matrix and write a h ik where h and k are hermitian matrices we show that if the minimal polynomial of the pencil xh yk has degree 3 then there is a unitary matrix u such that u 1au is block diagonal with blocks of size 3 3 or smaller this is a special case of a conjecture made by kippenhahn in 1951

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of n n complex matrices a b under the simultaneous similarity tat 1 tbt 1 roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set m 0n 2 r mn mn mn the set of n n complex valued matrices here r and are two positive integers then we construct a finite number of rational functions o1 os in the entries of a and b whose values are constant on all pairs similar in m n 2 r to a b the values of the functions oi a b i 1 s determine a finite number at most n 2 r of similarity classes in m n 2 r let sn be the subspace of complex symmetric matrices in mn for a b sn sn we consider the similarity class tatt tbtt where t ranges over all complex orthogonal matrices then the characteristic polynomial i a xb determines a finite number of similarity classes for almost all pairs a b sn sn