on the perturbation of the reduced minimum modulus of bounded linear operators let x and y be two banach spaces and let t x y be a bounded linear operator we study the perturbation problem of the reduced minimum modulus of t and in particular we correct a mistake in a recent paper

the expression of the generalized inverse of the perturbed operator under type i perturbation in hilbert spaces let h1 h2 be two hilbert spaces over the complex field c and let t h1 h2 be a bounded linear operator with the generalized inverse t let t t t be a bounded linear operator with t t 1 suppose that dim ker t dim ker t or r t r t 0 then t has the generalized inverse t i i t t 1 i t t i t i t t 1 1 1 tt 1t x i tt tt tt i tt 1 i tt ast 1tt i tt i 1 with t t 1 t t this result gives an analogue of theorem 3 9 of m z nashed generalized inverses and applications academic press new york 1976 in hilbert spaces

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

some equivalent conditions of stable perturbation of operators in hilbert spaces let h 1 h 2 be two hilbert spaces over the complex field c and let t email protected h 1 h 2 be two bounded linear operators with the m p generalized inverse t if r t r t 0 we say that t is the stable perturbation of t in this paper we give five equivalent conditions that make t being the stable perturbation of t under the assumption t email protected email protected email protected 1 these equivalent conditions generalize not only the notation of rank preserving perturbations of matrices but also the notation of acute perturbations of matrices as a result we obtain the following suppose that r t r t 0 and t email protected email protected email protected 1 email protected t t email protected email protected t email protected email protected t email protected email protected email protected this result generalizes corresponding result when h 1 h 2 are all finite dimensional

on the drazin inverse of the sum of two operators and its application to operator matrices abstract given two bounded linear operators f g on a banach space x such that g 2 f g f 2 0 we derive an explicit expression for the drazin inverse of f g for this purpose firstly we obtain a formula for the resolvent of an auxiliary operator matrix in the form m f i g f g from the provided representation of f g d several special cases are considered in particular we recover the case g f 0 studied by hartwig et al r e hartwig g wang y wei some additive results on drazin inverse linear algebra appl 322 2001 207 217 for matrices and by djordjevic and wei d s djordjevic y wei additive results for the generalized drazin inverse j aust math soc 73 1 2002 115 126 for operators finally we apply our results to obtain representations for the drazin inverse of operator matrices in the form m a b c d which are extensions of some cases given in the literature

multivariate smoothing spline functions given data z i g t i varepsilon i 1 leqq i leqq n where g is the unknown function the t i are known d dimensional variables in a domain omega and the varepsilon i are i

a unified framework for spline estimators this dissertation develops a uni ed framework to study the asymptotic properties of all periodic spline based estimators that is of regression penalized and smoothing splines the explicit form of the periodic demmler reinsch basis of general degree in terms of exponential splines allows to derive the exact expression for the equivalent kernel of all spline estimators simultaneously the corresponding bandwidth which drives the asymptotic behavior of spline estimators is shown to be a function of both the number of knots and the smoothing parameter a strategy for the optimal bandwidth selection is discussed

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

sharp norm estimations for moore penrose inverses of stable perturbations of hilbert c module operators it is well known that norm upper bound vert bar t dag vert leq frac vert t dag vert 1 vert t dag vert cdot vert delta t vert plays a fundamental role in stable resp rank preserving perturbation analysis for moore penrose inverses of hilbert space operators resp matrices in this paper in the general setting of hilbert c module operators we provide a new approach to the study of norm upper bounds of bar t dag and obtain a sharp estimation for vert bar t dag vert this obtained estimation is applied to the study of the linear least squares problem and of perturbation analysis for the schur complement of a positive semidefinite operator matrix some new norm upper bounds for schur complements of positive semidefinite operator matrices are derived

the role of pseudo data for robust smoothing with application to wavelet regression summary we propose a robust curve and surface estimator based on m type estimators and penalty based smoothing this approach also includes an application to wavelet regression the concept of pseudo data a transformation of the robust additive model to the one with bounded errors is used to derive some theoretical properties and also motivate a computational algorithm the resulting algorithm termed the es algorithm is computationally fast and provides a simple way of choosing the amount of smoothing moreover it is easily described straightforwardly implemented and can be extended to other wavelet regression settings such as irregularly spaced data and image denoising results from a simulation study and real data examples demonstrate the promising empirical properties of the proposed approach

on stable perturbations for outer inverses of linear operators in banach spaces abstract in this paper we investigate stable perturbations and their characterizations for various types of outer inverses such as generalized 2 3 2 4 2 5 1 2 3 1 2 4 moore penrose group drazin and generalized drazin inverses some known results are improved and extended

on the number of invariant polynomials of the matrix xax 1 b abstract let f be a field if a is an n n matrix over f we denote by i a the number of invariant polynomials of a different from 1 we shall prove that if a b are n n matrices over f and t 1 n then i a i b n t if and only if there exists a nonsingular matrix x over f such that i xax 1 b t except in a few cases

an improvement on the perturbation of the group inverse and oblique projection the perturbations of the group inverse a and oblique projection aa of a square matrix a have been previously studied under certain assumptions on the matrix a and a perturbation matrix e upper bounds for b bb b a a and bb aa aa where b a e are given in the literature recently upper bounds for the general case have been published by y wei appl math comp 98 1999 29 however the special cases in the literature and the continuity of the group inverse do not follow from the general upper bounds in this paper we derive new general upper bounds which not only cover all the special cases but also are sharper than wei s results such that the continuity of the group inverse directly follows a numerical example is given to illustrate the sharpness of the new general upper bounds

smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables longitudinal samples i e datasets with repeated measurements over time are common in biomedical and epidemiological studies such as clinical trials and cohort observational studies an exploratory tool for the analyses of such data is the varying coefficient model y t xt t t t where y t and x t x 0 t x k t t are the response and covariates at time t t 0 t k t t are smooth coefficient curves of t and t is a mean zero stochastic process a special case that is of particular interest in many situations is data with time dependent response and time independent covariates we propose in this article a componentwise smoothing spline method for estimating 0 t k t nonparametrically based on the previous varying coefficient model and a longitudinal sample of t y t x with time independent covariates x x 0 x k t from n independent subjects a leave one subject out cross validation is suggested to choose the smoothing parameters asymptotic properties of our

a class of grouped brunk estimators and penalized spline estimators for monotone regression we study a class of monotone univariate regression estimators we use b splines to approximate an underlying regression function and estimate spline coefficients based on grouped data we investigate asymptotic properties of two monotone estimators a grouped brunk estimator and a penalized monotone estimator these estimators are consistent at the boundary and their mean square errors achieve optimal convergence rates under suitable assumptions of the true regression function asymptotic distributions are developed and are shown to be independent of spline degrees and the number of knots simulation results and car data illustrate performance of the proposed estimators

finite dimensional approximation settings for infinite dimensional moore penrose inverses after a brief survey of basic results about finite dimensional approximation settings such as mathematical frameworks of various projection methods including the least squares method the dual least squares method and the galerkin method for infinite dimensional moore penrose inverses this paper proceeds to a detailed study from the following aspects for projection methods we investigate convergence and weak convergence of their approximation setting to develop a unified theory on projection methods for infinite dimensional moore penrose inverses this investigation yields a fundamental convergence theorem theorem 2 2 from which the criterion for convergence the criterion for weak convergence and the generalized dual least squares method are derived we also derive general results on the least squares method by which two flaws in groestch s results are corrected for nonprojection methods whose approximation setting is a more general framework we investigate weak perfect convergence of their approximation setting and provide a necessary and sufficient condition of such convergence holding theorem 3 2 several examples are proposed as counterexamples to illustrate the differences between some important concepts or as concrete algorithms to show how the present work can help to analyze their behavior

the number of invariant polynomials of a matrix with prescribed off diagonal blocks abstract let a be an n n matrix over an arbitrary field f of the form a a 1 2 a 1 2 a 2 1 a 2 2 where a1 1 fp p and a2 2 fq q and p q n we characterize the possible number of nontrivial invariant polynomials of a when the submatrices a1 2 and a2 1 are prescribed

perturbation theory for the generalized bott duffin inverse and its applications a perturbation theory for the bott duffin inverse a l of a with respect to a subspace l is developed the perturbation bound for the solution of the constrained system ax y b email protected l email protected l is established where email protected m n c a subspace email protected c n and email protected c n meanwhile this paper shows the generalized bott duffin condition number k g b d a email protected email protected email protected a l to be minimum in the inequality of error analysis

perturbations and expressions for generalized inverses in banach spaces and moore penrose inverses in hilbert spaces of closed linear operators abstract the problems of perturbation and expression for the generalized inverses of closed linear operators in banach spaces and for the moore penrose inverses of closed linear operators in hilbert spaces are studied we first provide some stability characterizations of generalized inverses of closed linear operators under t bounded perturbation in banach spaces which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression t i tt 1 utilizing these results we investigate the expression for the moore penrose inverse of the perturbed operator in hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation an explicit representation for the moore penrose inverse of the perturbation is also given moreover we give an equivalent condition for the moore penrose inverse to have the simplest expression t i tt 1 the results obtained in this paper extend and improve many recent results in this area

expression for the perturbation of the weighted moore penrose inverse abstract we consider the perturbation formula for the weighted moore penrose inverse of a rectangular matrix and give an explicit expression for the weighted moore penrose inverse of a perturbed matrix under the weakest rank condition this explicit expression extends the earlier work of several authors

the expression of the generalized bott duffin inverse and its perturbation theory let l be a subspace of c n and p l be the orthogonal projection of c n onto l then for the nxn matrix a the generalized bott duffin b d inverse a l is given by a l p l ap l i p l in this paper we will give an expression of a l in the rest of the part the perturbation analysis of a l and the least squares solution of the generalized b d equation ax y b for x l and y l will be established under certain small perturbation of p lap l

new perturbation results on pseudo inverses of linear operators in banach spaces abstract let x and y be banach spaces let t x y be a bounded linear operator with closed range and let s x y be another bounded linear operator we study conditions on s t that guarantee the closeness of the range of s and obtain some new bounds on the pseudo inverse of s

on the continuity of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range we study the continuity problem of the generalized inverse of t and related least squares solutions to the operator equation tx y

matrices with prescribed eigenvalues and principal submatrices abstract we give a necessary and sufficient condition for the existence of a square matrix with prescribed eigenvalues and prescribed complementary principal blocks

on the perturbation of the least squares solutions in hilbert spaces abstract let h1 and h2 be hilbert spaces and let t h1 h2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

on perturbations for oblique projection generalized inverses of closed linear operators in banach spaces the main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces we provide a new stability characterization of oblique projection generalized inverses of closed linear operators under t bounded perturbations which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space

matrices with prescribed nonprincipal blocks and the trace function abstract let a and b be fixed matrices we discuss the existence of matrices u and v such that vau 1 ubv 1 is nonscalar and simultaneously has a prescribed trace

uniform convergence of regularization methods for linear ill posed problems abstract after a general discussion about convergence and convergence rates for regularization methods in banach spaces we present a general method that can be used to modify regularization methods in l2 in such a way that uniform convergence which is often preferred in concrete applications to just l2 convergence is obtained we prove results about convergence rates in the uniform norm and discuss questions of parameter choice the theoretical results will be supported by numerical examples which indicate that although the results are asymptotic in character they are of some relevance also for actual computations

the characterizations of the stable perturbation of a closed operator by a linear operator in banach spaces in this paper we investigate the invertibility of iy tt when t is a closed operator from x to y with a generalized inverse t and t is a linear operator whose domain contains d t and range is contained in d t the characterizations of the stable perturbation t t of t by t in banach spaces are obtained the results extend the main results in 7 8

perturbation for a pair of oblique projectors aa mn and bb mn abstract by using the method of generalized singular values decomposition we establish a canonical theorem for a pair of oblique projectors aa mn and bb mn give some new results for perturbations of aa mn and bb mn where a mn is the weighted moore penrose inverse of matrix a m and n are two hermitian positive definite matrices and b is a matrix obtained from a perturbation of a our work can be seen as the extended results for orthogonal projectors aa and bb some numerical examples are given to illustrate the results

approximative compactness and continuity of the set valued metric generalized inverse in banach spaces abstract in this paper continuity of the set valued metric generalized inverse t in approximatively compact banach spaces is investigated by means of the methods of geometry of banach spaces necessary and sufficient conditions for upper semicontinuity continuity for the set valued metric generalized inverses t are given moreover authors also prove that if x is a nearly dentable space and h is a hyperplane of x then h is approximatively compact iff p h x is compact for any x x

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

smoothing splines with varying smoothing parameter this paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty the penalty is a function of the design points in order to accommodate local behaviour of the regression function we show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator and that the equivalent kernel is spatially dependent the equivalent kernels for traditional smoothing splines are a special case of this general solution with the aid of the green s function for a two point boundary value problem explicit forms of the asymptotic mean and variance are obtained for any interior point thus the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error simulation results and an application illustrate the performance of the proposed estimator copyright 2013 oxford university press

operators with equal projections related to their generalized inverses in this article we characterize operators on banach spaces which have the same projections related to their outer or inner generalized inverses as corollaries we obtain well known results for the drazin inverse of bounded operators

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

perturbation analysis of the least squares solution in hilbert spaces abstract let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we give a complete description of the perturbation analysis for the least squares solution to the operator equation t x y where x h 1 y h 2

perturbation analysis of generalized inverses of linear operators in banach spaces abstract let x 1 x 2 be two banach spaces and t x 1 x 2 be a bounded linear operator with a bounded generalized inverse t we study the perturbation problem of generalized inverse t and provide two new stability characterizations of generalized inverse we also give a continuity characterization of moore penrose inverse in hilbert spaces our results are new even in the case of matrices

perturbation of least squares problem in hilbert spaces let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

a comparison of a spline estimate to its equivalent kernel estimate it has been observed that to a smoothing spline operator there corresponds an equivalent kernel operator these two operators have been compared in a variety of norms cox 1984 silverman 1984 in this paper we refine the existing bounds for the particular case of the spline estimator considered in rice and rosenblatt 1983 and its corresponding equivalent kernel estimator we obtain detailed asymptotic expressions for the bias and covariance functions of the two estimates and provide rates of convergence direct comparison then shows that the two estimates are similar they may differ somewhat in their higher order boundary behavior

the number of nonconstant invariant polynomials of matrices with several prescribed blocks abstract consider a n n matrix partitioned into k k blocks c c i j where c 1 1 c k k are square this paper studies the possible numbers of nonconstant invariant polynomials of c when a diagonal of blocks c i j is fixed and the others vary

on the error estimate for the projection of a point onto a linear manifold abstract suppose that the linear system a h x b is consistent the problem of finding the solution x to min a h x b 1 2 x p 2 2 is studied where a c n m b c m x p c n and a b p are given when a b and p are perturbed an error bound for the solution x is obtained roundoff error bounds for the singular value decomposition method the column pivoting method and the rank revealing method are obtained the results in this paper extend those of arioli laratta and menchi to the case when a is not of full rank and either m n or m n

estimation of monotone functions via p splines a constrained dynamical optimization approach estimation of monotone functions has broad applications in statistics engineering and science this paper addresses asymptotic behaviors of monotone penalized spline estimators using constrained dynamical optimization techniques the underlying regression function is approximated by a b spline of an arbitrary degree subject to the first order difference penalty the optimality conditions for spline coefficients give rise to a size dependent complementarity problem as a key technical result of the paper the uniform lipschitz property of optimal spline coefficients is established by exploiting piecewise linear and polyhedral theories this property forms a cornerstone for stochastic boundedness uniform convergence and boundary consistency of the monotone estimator the estimator is then approximated by a solution of a differential equation subject to boundary conditions this allows the estimator to be represented by a kernel regression estimator defined by a related green s function of an ode the asymptotic normality is established at interior points via the green s function the convergence rate is shown to be independent of spline degrees and the number of knots does not affect asymptotic distribution provided that it tends to infinity fast enough

on the semi continuity of generalized inverses in banach algebras abstract the main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a banach algebra with identity we first obtain a sufficient and necessary condition for the lower semi continuity of reflexive generalized inverses as set valued mappings based on this result we characterize the continuity of the moore penrose inverse in a c algebra and therefore derive some new and well known criteria in operator theory

a generalization of a classic theorem in the perturbation theory for linear operators let x and y be banach spaces and let t x y be a bounded linear operator with closed range in this paper we extend a classic perturbation result from the invertible case to the general case for the consistent operator equation tx y

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

condition number related with generalized inverse a t s 2 and constrained linear systems in this paper we introduce a pq norm and discuss the condition number of the generalized inverse at s 2 we get some results on the condition numbers with respect to at s 2 and constrained linear systems ax b x t which are similar to the regular inverse a 1 and nonsingular linear systems ax b finally we investigate the structured perturbation of the generalized inverse at s 2

on the number of invariant polynomials of partially prescribed matrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 a 1 2 a 2 2 are prescribed

perturbation analysis for the operator equationtx bin banach spaces letx1 x2be two banach spaces over the complex fieldcand lett x1 x2be a bounded linear operator with the generalized inverset lett t tbe a bounded linear operator with t t 1 suppose thatdimkert dimkert orr t kert 0 thenthas the generalized inverset i t t 1t with t t 1 t t this result generalizes theorem 3 9 ofna using this result we give some results of the perturbation analysis for the operator equationtx b

perturbation and expression for inner inverses in banach spaces and its applications abstract it is well known that if t is invertible and t 1 is its inverse then t 1 i t t t 1 1 is the inverse of t for t t t 1 1 the main purpose of this paper is to study the following problem if t is an inner inverse of t and t t t 1 is t i t t t 1 an inner inverse of t if the answer is no when is t i t t t 1 an inner inverse of t in this paper we give a complete answer to these problems and provide some characterizations for t i t t t 1 to be an inner inverse of t in banach spaces utilizing these results we investigate similar problems for the generalized inverse 1 3 inverse 1 4 inverse 1 5 inverse 1 2 3 inverse 1 2 4 inverse moore penrose inverse group inverse and drazin inverse the results obtained in this paper extend and improve many recent results in this area

the number of invariant polynomials of a matrix with prescribed complementary principal submatrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 and a 2 2 are prescribed

on the number of invariant factors of partially prescribed matrices and control theory abstract let a and b be matrices of sizes m m and m u respectively over an infinite field let i a b be the number of invariant factors of xi a b different from 1 this paper gives a necessary and sufficient condition for the existence of a matrix a b with a prescribed submatrix on the right lower corner such that i a b in particular this paper studies the possibility of a linear system t a t b t being completely controllable when a submatrix on the right lower corner of a b is unknown and all the other entries are prescribed

perturbation analysis for the reduced minimum modulus of bounded linear operator in banach spaces let x and y be two banach spaces over the complex field c and let t x y be a bounded linear operator with the generalized inverse t let t t dt be a bounded linear operator with t dt 1 2 let c t denote the reduced minimum modulus of t we first establish the upper semi continuity theorem for c t furthermore if dimkert dimkert or r t kert 0 then c t is continuous under the operator norm also we have a corollary in hilbert spaces which improves the result in linear algebra appl 262 1997 229 242 theorem 4 1

matrices with prescribed submatrices and number of invariant polynomials abstract we give necessary and sufficient conditions for the existence of a matrix a b over a field where a is a square matrix such that a b has a prescribed submatrix and the pencil xi a b has a prescribed number of invariant factors different from 1 as a corollary we give a necessary and sufficient condition for the existence of a completely controllable pair a b such that a b has a prescribed submatrix

a class of outer generalized inverses abstract in any semigroup or semigroup s it is shown that the moore penrose inverse y a the author s pseudo inverse y a chipman s weighted inverse and the bott duffin inverse are all special cases of the more general class of b c inverses y s satisfying y bsy ysc yab b and cay c these b c inverses always satisfy yay y are always unique when they exist and exist if and only if b scab and c cabs in which case under the partial order m of mitsch y is also the unique m greatest element of the set x a x a b c x x s xax x and x bsx xsc and the unique m least element of z a z a b c z z s zaz z zab b and caz c the above all holds in arbitrary semigroups s hence in particular in any associative ring r for any complex n n matrices a b c an efficient uniform procedure is given to compute the b c inverse of a whenever it exists in the ring case a r is called weakly invertible if there exist b c r satisfying 1 b 1 a r 1 c r 1 a such that a has a b c inverse y satisfying ay ya and it is shown that a is weakly invertible if and only if a is strongly clean in the sense of nicholson i e a u e for some unit u and idempotent e with eu ue

matrix completion problems abstract throughout the last decades several results have been published in the area of the so called matrix completion problems in this paper we survey several results in this field in particular we describe the possible eigenvalues the characteristic polynomial the invariant polynomials or the number of nontrivial invariant polynomials of a square matrix over a field when some of its entries are prescribed and the others vary finally we present our contribution generalizing some of the previous cases to an n n matrix partitioned into k k blocks with entries in a field when some of its blocks are prescribed and the others vary

perturbation analysis for oblique projection generalized inverses of closed linear operators in banach spaces in this paper we study the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces by the method of the perturbation analysis of linear operators we obtain an explicit perturbation theorem and error estimates for the oblique projection generalized inverse of closed linear operators under the t bounded perturbation which extend the known results on the perturbation of the oblique projection generalized inverse of bounded linear operators in banach spaces

some remarks on matrix norms condition numbers and error estimates for linear equations abstract inequalities between some norms of rectangular matrices and the corresponding relationships between condition numbers are established and clearly arranged in two simple diagrams furthermore some well known error estimates for linear equations in the nonsingular case are shown to be valid for all submultiplicative matrix norms the new proofs work without using infinite series

the perturbation of consistent least squares problems abstract suppose that the linear system ax b is consistent and a a a b b b are perturbed from a b respectively where a and are m n matrices b and b are m dimensional vectors let a f e a 2 b 2 e b 2 and a 2 a 2 be the condition number of a it is shown that if e 1 then the least squares solution x of a x b is close to a solution x of ax b even when a is not of full rank and rank rank a

choosing a range for the amount of smoothing in nonparametric regression abstract most nonparametric regression estimates smooth the observed data in the process of generating a suitable estimate the resulting curve estimate is sensitive to the amount of smoothing introduced and it is important to examine the estimate over a range of smoothing choices an objective way to specify such a range is to estimate the optimal value of the smoothing parameter bandwidth with respect to some loss function and then report a confidence interval for this parameter estimate this interval suggests examining three regression estimates an undersmoothed curve evaluated at the lower endpoint of the confidence interval an oversmoothed curve evaluated at the upper endpoint and also an optimal curve chosen by the data based method this article describes two strategies for constructing such confidence intervals using asymptotic approximations and simulation techniques suppose that data are observed from the model yk f tk ek 1 k n where f is a function that is twice differentia

unified theorems on completions of matrix pencils abstract we generalize and unify some theorems concerned with the existence of a pencil with a prescribed subpencil and prescribed strict equivalence class

matrices with prescribed characteristic polynomial and principal blocks ii abstract in this paper we present some new sufficient conditions for the existence of a square matrix with prescribed characteristic polynomial and two complementary principal submatrices

eigenvalues of matrices with several prescribed blocks ii abstract a previous paper by d hershkowitz linear and multilinear algebra 14 1983 315 342 described the possible eigenvalues of an n n matrix when 2n 3 entries are fixed and the others vary this paper describes the possible eigenvalues of a pk pk matrix partitioned into k k blocks of size p p when 2k 3 blocks are fixed and the others vary

on the number of invariant polynomials of matrix commutators we study the possible numbers of noneonstant invariant polynomials of the matrix commutator xa ax when x varies elsevier science inc 1997

on the perturbation of the group generalized inverse for a class of bounded operators in banach spaces abstract given a bounded operator a on a banach space x with drazin inverse a d and index r we study the class of group invertible bounded operators b such that i a d b a is invertible and r b n a r 0 we show that they can be written with respect to the decomposition x r a r n a r as a matrix operator b b 1 b 12 b 21 b 21 b 1 1 b 12 where b 1 and b 1 2 b 12 b 21 are invertible several characterizations of the perturbed operators are established extending matrix results we analyze the perturbation of the drazin inverse and we provide explicit upper bounds of b a d and b b a d a we obtain a result on the continuity of the group inverse for operators on banach spaces

some new perturbation theorems for generalized inverses of linear operators in banach spaces abstract et x y be banach spaces and let t b x y be a linear operator with closed range r t we consider the perturbation problem for the bounded outer inverse the bounded 2 3 inverse and the bounded generalized inverse of t with respect to the projectors some new perturbation theorems are given by the generalized neumman lemma and the concept of stable perturbations in banach spaces