some schur complement identities and applications to matrix extrapolation methods the schur complement has many applications in linear algebra and in numerical analysis schur complement identities have been used for implementing some scalar and vector extrapolation methods in this paper we define some matrix sequence transformations and give two new schur complement identities that will be used to derive algorithms for implementing these transformations

spectral shorted matrices abstract given a n n positive semidefinite matrix a and a subspace s of c n s a denotes the shorted matrix of a to s we consider the notion of spectral shorted matrix s a lim m s a m 1 m wecompletely characterize this martix in terms of s and the spectrum and the eigenspaces of a we show the relation of this notion with the spectral order of matrices and the kolmogorov s complexity of a to a vector c n

geometry structures of the generalized inverses of block two by two matrices abstract in this paper we study the invertibility and the moore penrose invertibility of 2 2 block matrix with schur complement having closed range the necessary and sufficient conditions for the existence as well as the expressions for the inverses and the moore penrose inverses are obtained

on inverse positivity of sub direct sums of matrices abstract in this note the authors consider the problem of inverse positivity of k subdirect sum of matrices the main results provide a solution to an open problem posed recently

efficient iteration schemes for anisotropic hyperelasto plasticity numerical issues arising when integrating hyperelasto plastic constitutive equations with elastic anisotropy stemming from anisotropic damage and plastic anisotropy as represented by kinematic hardening are discussed in particular solution algorithms for the corresponding non linear system of equations due to implicit integration algorithms are addressed it is shown that algorithms like staggered iteration and quasi newton techniques are superior to a pure newton technique when the cpu time is compared however the drawback of the staggered and the quasi newton technique is that rather small time steps must be taken to ensure convergence which can be of importance when applying complex constitutive models in a finite element programme copyright 2005 john wiley sons ltd

matrix semigroups with constant spectral radius abstract multiplicative matrix semigroups with constant spectral radius c s r are studied and applied to several problems of algebra combinatorics functional equations and dynamical systems we show that all such semigroups are characterized by means of irreducible ones each irreducible c s r semigroup defines walks on euclidean sphere all its nonsingular elements are similar in the same basis to orthogonal we classify all nonnegative c s r semigroups and arbitrary low dimensional semigroups for higher dimensions we describe five classes and leave an open problem on completeness of that list the problem of algorithmic recognition of c s r property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones

a generalization of a a 1 i abstract a matrix inequality is obtained in an elementary way for the schur product of two positive definite matrices the inequality generalizes a well known result due to fiedler which asserts that if a is a positive definite matrix then a a 1 i is a positive semidefinite matrix where denotes the schur multiplication

remarks on the schur complement abstract first a generalization of the schur determinantal formula is given using properties of quasidirect sums of matrices a new characterization of the schur complement is proved

completing a symmetric 2 2 block matrix and its inverse abstract we consider the following completion problems suppose n1 n2 are nonnegative integers such that n 1 n 2 n 0 let a11 a12 a21 b22 be matrices with dimensions n1 n1 n1 n2 n2 n1 and n2 n2 respectively we determine necessary and sufficient conditions so that there exists an n2 n2 matrix a22 such that a a 11 a 21 a 12 a 22 and i a is nonsingular and symmetric and b22 is the lower right block of a partitioning of a 1 ii a is symmetric positive definite and b22 is the lower right block of a partitioning of a 1

penrose demosaicking the penrose pixel layout an aperiodic pixel layout in rhombus penrose tiling has been shown to substantially outperform the existing square pixel layout in super resolution however it was tested only on grayscale images to study its performance on color images we have to reconstruct regular color images from penrose raw images i e images with only one color component at each penrose pixel resulting in the problem of demosaicking from penrose pixels penrose demosaicking is more difficult than regular demosaicking because none of the color components of the reconstructed regular color images are available therefore most of the traditional demosaicking methods do not apply we develop a sparse representation based method for penrose demosaicking extensive experiments show that penrose pixel layout outperforms regular pixel layouts in terms of both perceptual evaluation and s cielab the penrose pixel layout is unique among all irregular layouts because it is uniformly three colorable and it has only two pixel shapes thick and thin rhombi making its manufacturing relatively easy

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

non euclidean geometrical aspects of the schur and levinson szego algorithms in this paper we address non euclidean geometrical aspects of the schur and levinson szego algorithms we first show that the lobachevski geometry is by construction one natural geometrical environment of these algorithms since they necessarily make use of automorphisms of the unit disk we next consider the algorithms in the particular context of their application to linear prediction then the schur resp levinson szego algorithm receives a direct resp polar spherical trigonometry st interpretation which is a new feature of the classical duality of both algorithms

computer assistance for discovering formulas in system engineering and operator theory abstract the objective of this paper is two fold first we present a methodology for using a combination of computer assistance and human intervention to discover highly algebraic theorems in operator matrix and linear systems engineering theory since the methodology allows limited human intervention it is slightly less rigid than an algorithm we call it a strategy the second objective is to illustrate the methodology by deriving four theorems the presentation of the methodology is carried out in three steps the first step is introducing an abstraction of the methodology which we call an idealized strategy this abstraction facilitates a high level discussion of the ideas involved idealized strategies cannot be implemented on a computer the second and third steps introduce approximations of these abstractions which we call prestrategy and strategy respectively a strategy is more general than a prestrategy and in fact every prestrategy is a strategy the above mentioned approximations are implemented on a computer we stress that since there is a computer implementation the reader can use these techniques to attack their own algebra problems thus the paper might be of both practical and theoretical interest to analysts engineers and algebraists now we give the idea of a prestrategy a prestrategy relies almost entirely on two commands which we call ncprocess1 and ncprocess2 these two commands are sufficiently powerful so that in many cases when one applies them repeatedly to a complicated collection of equations they transform the collection of equations into an equivalent but substantially simpler collection of equations a loose description of a prestrategy applied to a list of equations is emsp 1 declare which variables are known and which are unknown at the beginning of a prestrategy the order in which the equations are listed is not important since ncprocess1 and ncprocess2 will reorder them so that the simplest ones appear first 2 apply ncprocess1 to the equations the output is a set of equations usually some in fewer unknowns than before carefully partitioned based upon which unknowns they contain 3 the user must select important equations especially any which solve for an unknown sayx when an equation is declared to be important or a variable is switched from being an unknown to being a known then the way in which ncprocess1 and ncprocess2 reorder the equations is modified 4 switchxto being known rather than unknown go to 2 above or stop when this procedure stops it hopefully gives the canonical necessary conditions for the original equations to have a solution as a final step we run ncprocess2 which aggressively eliminates redundant equations and partitions the output equations in a way which facilitates proving that the necessary conditions are also sufficient many classical theorems in analysis can be viewed in terms of solving a collection of equations we have found that this procedure actually discovers the classic theorem in a modest collection of classic cases involving factorization of engineering systems and matrix completion problems one might regard the question of which classical theorems in analysis can be proven with a strategy as an analog of classical euclidean geometry where a major question was what can be constructed with a compass and ruler here the goal is to determine which theorems in systems and operator theory could be discovered by repeatedly applying ncprocess1 and ncprocess2 or their successors and the human selection of equations which are important the major practical challenge addressed here is finding operations which when implemented in software present the user with crucial algebraic information about his problem while not overwhelming him with too much redundant information this paper consists of two parts a description of strategies a high level description of the algorithms a description of the applications to operator matrix and linear system engineering theory and a description of how one would use a strategy to discover four different theorems are presented in the first part of the paper thus one who seeks a conventional viewpoint for this rather unconventional paper might think of this as providing a unified proof of four different theorems the theorems were selected for their diverse proofs and because they are widely known so that many readers should be familiar with at least one of them the ncprocess commands use noncommutative grobner basis algorithms which have emerged in the last decade together with algorithms for removing redundant equations and a method for assisting a mathematician in writing a noncommutative polynomial as a composition of polynomials the reader needs to knownothingabout grobner basis to understand the first part of this paper descriptions involving the theory of grobner basis appear in the second part of the paper

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

on properties of generalized quadratic operators abstract in this paper we investigate the set p of generalized quadratic operators a satisfying the equation a 2 a p for all complex numbers and and for an idempotent operator p such that ap pa a furthermore the close relationship between the operator a p and the idempotent operator p are established and expressions for the inverse the moore penrose inverse and the drazin inverse of a p are given some related results are also obtained

the schur complement and the inverse m matrix problem abstract it is well known that if m is a nonnegative nonsingular inverse m matrix and if a is a nonsingular block in the upper left hand corner then the schur complement of a in m m a is an inverse m matrix the converse of this is generally false in this paper we give added restrictions on m or a to insure the converse and give some necessary and sufficient conditions for hmh 1 where h i i to be an m matrix

complementable and almost definite matrices abstract in recent papers 1 and 5 m complementable matrices were defined by t ando and almost definite matrices by duffin and morley in this paper additional characterizations are given for these sets of matrices making use of the generalized inverse also rank properties and generalized inverses of these matrices are discussed

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

on the semigroup property for some structured iterations abstract nonlinear matrix equations play a crucial role in science and engineering problems however solutions of nonlinear matrix equations cannot in general be given analytically one standard way of solving nonlinear matrix equations is to apply the fixed point iteration with usually only the linear convergence rate to advance the existing methods we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order we realize our way by starting with examples of solving the scalar equations and also connect this method with some well known equations including but not limited to the stein matrix equation the generalized eigenvalue problem the generalized nonlinear matrix equation the discrete time algebraic riccati equations to express the capacity of this method

expression of the drazin and mp inverse of partitioned matrix and quotient identity of generalized schur complement in this paper we give representations of the drazin and mp inverse of a 2x2 block matrix and quotient identities for the generalized schur complement of a partitioned 3x3 matrix under conditions different than those used in recent papers on the subject we present numerical examples to illustrate our results

generalized inverses of partitioned matrices useful in statistical applications abstract generalized inverses of a partitioned matrix are characterized under some rank conditions on the block matrices in the partitions

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

a maximum principle for the stability analysis of positive bilinear control systems with applications to positive linear switched systems we consider a continuous time bilinear control system bcs with metzler matrices each entry in the transition matrix of such a system is nonnegative making the positive orthant an invariant set of the dynamics motivated by the stability analysis of positive linear switched systems plsss we define a control as optimal if for a fixed final time it maximizes the spectral radius of the transition matrix our main result is a first order necessary condition for optimality in the form of a maximum principle mp the proof of this mp combines the standard needle variation with a basic result from the perron frobenius theory of nonnegative matrices we describe several applications of this mp to the stability analysis of plsss under arbitrary switching

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

an upper bound on the number of linear relations identified from noisy data by the frisch scheme abstract the frisch scheme is one of many approaches to the identification of linear relations from noisy data solution of the associated matrix optimization problem leads to the identification of a maximal number of such relations to date no general solution procedure is known here an upper bound on the maximal number of relations that can be identified is presented the upper bound is derived using the theory of schur complements in conjunction with a result for the case where only one relation can be identified the possibility of developing a completely general solution procedure using these ideas is also briefly discussed

on disjoint range operators in a hilbert space for a bounded linear operator m in a hilbert space h various relations among the ranges r m r m r m m and the null spaces n m n m are considered from the point of view of their relations to the known classes of operators such as ep co ep weak ep gp dr or sr particular attention is paid to the range projectors of the operators m m and some further characteristics of these projectors are derived as well

explicit and provably stable spatiotemporal fdtd refinement abstract in this paper we introduce an explicit and provably conditionally stable finite difference time domain fdtd algorithm for maxwell s equations with local refinement in both the spatial discretization length and in the time step spatiotemporal refinement this enables local spatial refinement with a locally reduced time step

when do several linear operators share an invariant cone abstract we establish a criterion for a finite family of matrices to possess a common invariant cone the criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family in spite of theoretical simplicity the practical use of the criterion may be difficult we show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable corollaries of the criterion which give sufficient and necessary conditions are derived finally we introduce a co directional number of several matrices we prove that this parameter is close to zero iff there is a small perturbation of matrices after which they get an invariant cone an algorithm for its computation is presented

the extreme points of a set of positive semidefinite operators abstract we determine the extreme points and exposed points of the convex set of positive semidefinite operators less than a given positive semidefinite operator a on a finite dimensional space the extreme and exposed points are the same and are obtained by shorting the operator a to all subspaces of the space on which it is defined the extreme and exposed subsets are the intervals between the shorted operators corresponding to two subspaces one contained in the other

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

gravitational instantons of type d k and a generalization of the gibbons hawking ansatz we describe a quaternionic based ansatz generalizing the gibbons hawking ansatz to a class of hyperkahler metrics with hidden symmetries we then apply it to obtain explicit expressions for gravitational instanton metrics of type d k

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

irreducible semigroups with multiplicative spectral radius abstract irreducible multiplicative semigroups of finite dimensional linear operators with some additional properties are studied each of the properties submultiplicativity of the spectral radius its multiplicativity or is constancy as well as a certain property called the rota condition implies that every member of such a semigroup is except for a scalar coefficient similar to the direct sum of an isometry and a nilpotent operator

simultaneous diagonalization of rectangular matrices abstract a matrix d is said to be diagonal if its i j th element is null whenever i and j are unequal for a set a of matrices a of the same order the paper gives necessary and sufficient conditions for nonsingular matrices s and t to exist such that sa t d is diagonal for each matrix a in the set

parameter identification in constitutive models via optimization with a posteriori error control in this paper we outline a computational technique for the calibration of macroscopic constitutive laws with automatic error control in the most general situation the state variables of the constitutive law as well as the material parameters are spatially non homogeneous the experimental observations are given in space time based on an appropriate dual problem we compute a posteriori the discretization error contributions from approximations of the parameter state and costate elds in space time for an arbitrarily chosen goal oriented error measure of engineering signi cance such a measure can be used in an adaptive strategy not discussed in this paper to meet a prede ned error tolerance an important observation is that the jacobian matrix associated with the resulting newton method is used in principle in solving the dual problem rather than treating the jacobian in a monolithic fashion we utilize a sequential solution strategy whereby the fe topology of the discretized state problem is used repeatedly moreover the proposed solution strategy lends itself naturally to the computation of rst and second order sensitivities which are obtained with little extra computational effort numerical results are given for the prototype model of con ned aquifer ow with spatially non homogeneous permeability the ef ciency of the optimization strategy and the effectivity of the error computation are assessed

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

the inverse of any two by two nonsingular partitioned matrix and three matrix inverse completion problems a formula for the inverse of any nonsingular matrix partitioned into two by two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix the formula is then applied to three matrix inverse completion problems to obtain their complete solutions

new vector sequence transformations many numerical methods produce sequences of vectors converging to the solution of a problem when the convergence is slow the sequence can be transformed into a new vector sequence which under some assumptions converges faster to the same limit the construction of a sequence transformation is based on its kernel that is the set of sequences which are transformed into a constant sequence in this paper new vector sequence transformations are built from kernels which extent those of the most general transformations known so far

generalized schur complements and oblique projections let s be a closed subspace of a hilbert space h and a a bounded linear selfadjoint operator on h in this note we show that the existence of a selfadjoint projections with range s is related to some properties of shorted operators schur complements in ando s generalization of the classical concept and compressions 2001 elsevier science inc all rights reserved

feedback strategies for discrete time linear quadratic two player descriptor games this paper deals with discrete time linear quadratic descriptor two player games the two frameworks of nash and stackelberg strategies with feedback information structure are considered the aim is to provide sufficient conditions ensuring for any initial state the existence and the uniqueness of a tri trajectory gathering the state and the controls of the two players which reaches the equilibrium the provided results are mainly based on a discrete time matrix block formulation the provided algorithms consist in an iterative backward in time procedure issued from dynamic programming numerical examples illustrate this approach and the particular case of explicit systems is recovered for both kinds of strategies

an explicit time integration method for structural dynamics using septuple b spline functions in this paper an explicit time integration method with three parameters is proposed for structural dynamics using periodic septuple b spline interpolation polynomial functions in this way by use of septuple bsplines the authors have proceeded to solve the de of motion governing a single dof system and later the presented method has been generalized for a multiple dof system in the proposed method a direct recursive formula for response of the system was formulated on the basis of septuple b spline interpolation approximation in terms of the specific requirements of this proposed method two initialization approaches are given for initial calculation one is called direct initialization and the other is indirect initialization the stability analysis of the proposed method illustrates that by use of adjustable parameters a highfrequency response can be damped out without inducing excessive algorithmic damping in important low frequency modes the computational accuracy and efficiency of the proposed method is demonstrated with three numerical examples and the results from the proposed method are compared with those from some of the existent numerical methods such as the newmark and wilsonmethods the compared results show that the proposed method has high accuracy with low time consumption copyright 2013 john wiley sons ltd

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

approximation by matrices positive semidefinite on a subspace abstract we obtain the best approximation to a matrix by matrices positive semidefinite on a subspace as a by product we present two new characterizations of euclidean distance matrices

general expressions for the moore penrose inverse of a 2 2 block matrix abstract the moore penrose inverse of a 2 2 block matrix m m a c b d is discussed general expressions for the moore penrose inverse for the block matrix m in terms of the individual blocks a b c d are delivered without any restrictions imposed under some conditions the moore penrose inverse can be simplified the results extend earlier work by various authors

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

on matrix semigroups bounded above and below abstract an irreducible norm closed semigroup of complex matrices is simultaneously similar to a semigroup of partial isometries if and only if a the norms of all nonzero members of it are uniformly bounded above and below and b its idempotents commute this is a generalization of the well known result on bounded groups

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

representations of generalized inverses of partitioned matrix involving schur complement in this article we consider some representations of 1 3 1 4 1 2 3 and 1 2 4 inverses of the partitioned matrix m which are equivalent to some rank additivity conditions we present the application of these results to generalizations of the sherman morrison woodbury type formulae