ray solvable linear systems and ray s2ns matrices abstract ray solvable linear systems and ray s 2 ns matrices are complex generalizations of the sign solvable linear systems and s 2 ns matrices we use the determinantal ray unique matrices instead of ray nonsingular matrices as a generalization of sns matrices to generalize some fundamental results of s 2 ns matrices from the real case to complex case such as the graph theoretical characterization the inverse ray patterns and the upper bound of the number of nonzero entries of s 2 ns matrices the well known characterization of the sign solvable linear systems in terms of the l matrices and s matrices is also generalized to ray solvable linear systems and the relationships between the ray s matrices and real s matrices are investigated some examples are also given to illustrate that some results such as the characterization of the sign inconsistent linear systems do not carry over to the complex case

group inverse for the block matrices with an invertible subblock let m abcd a is square be a square block matrix with an invertible subblock over a skew field k in this paper we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for m under some conditions

sign patterns that require almost unique rank abstract a sign pattern matrix is a matrix whose entries are from the set 0 for a real matrix b sgn b is the sign pattern matrix obtained by replacing each positive respectively negative zero entry of b by respectively 0 for a sign pattern matrix a the sign pattern class of a denoted q a is defined as b sgn b a the minimum rank mr a maximum rank mr a of a sign pattern matrix a is the minimum maximum of the ranks of the real matrices in q a several results concerning sign patterns a that require almost unique rank that is to say the sign patterns a such that mr a mr a 1 are established and are extended to sign patterns a for which the spread is d mr a mr a a complete characterization of the sign patterns that require almost unique rank is obtained

some results on the drazin invertibility and idempotents in this paper several equivalent conditions for the drazin invertibility of the sum difference and the product of idempotents in banach algebra are established

a unified approach to krylov subspace methods for the drazin inverse solution of singular nonsymmetric linear systems abstract consider the linear system ax b where a c n n is a singular matrix in the present work we propose a general framework within which krylov subspace methods for drazin inverse solution of this system can be derived in a convenient way the krylov subspace methods known to us to date treat only the cases in which a is hermitian and its index ind a is unity necessarily in the present work a is not required to be hermitian it can have any type of spectrum and ind a is arbitrary we show that as is the case with nonsingular systems the krylov subspace methods developed here terminate in a finite number of steps that is at most n ind a for one of the methods derived here we also provide an analysis by which we are able to bound the errors the relevant bounds decreasing with increasing dimension of the krylov subspaces involved the results of this paper are applicable to consistent systems as well as to inconsistent ones an interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard krylov subspace approach to nonsingular systems indeed our approach here reduces to that relevant for nonsingular systems upon setting ind a 0 everywhere

convergence and quotient convergence of iterative methods for solving singular linear equations with index one singular systems with index one arise in many applications such as markov chain modelling in this paper we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one we give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems next we show that for the stationary iterative method the convergence and the quotient convergence are equivalent

characterizations iterative method sign pattern and perturbation analysis for the dmp inverse with its applications abstract this paper is a study on main properties and characterizations of the dmp inverse also corresponding representations and computational procedures are derived particularly an upgrade of the bordering method to the case of the dmp inverse is considered as well as applications of the dmp inverse in solving singular linear systems also the dmp inverse of an upper block triangular matrix and its sign pattern are investigated furthermore several computational procedures aimed to approximating the dmp inverse are developed implemented and tested the main of them are the limit representation a revised successive matrix squaring iterations and the gradient neural network gnn dynamical system finally some perturbation bounds and continuity of the dmp inverse are studied

further results on the representation of the drazin inverse of a 2 2 block matrix abstract the aim of this paper is to establish an explicit representation of the drazin inverse of a 2 2 block matrix m a b c d where a and d are square matrices in terms of the drazin inverses of a and d first we give a formula for the drazin inverse of m with the blocks satisfying the conditions bc 0 bdc 0 and bd 2 0 further we derive representations of the drazin inverse of m under less restrictive conditions than those found in the current literature

some block matrices with signed drazin inverses abstract the sign pattern of a real matrix m is the 0 1 1 matrix obtained from m by replacing each entry by its sign let q m be the set of real matrices with the same sign pattern as m for any m q m if the drazin inverses of m and m have the same sign pattern then m is said to have signed drazin inverse in this paper we give a complete characterization for a class of anti triangular matrices with signed drazin inverse and present a complete characterization for sign symmetric bipartite matrices with signed drazin inverse

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

a note on the group inverse of some 2 2 block matrices over skew fields abstract let k be a skew field and km n be the set of all m n matrices over k in this paper we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix ax yb a b 0 under some conditions where a y km n x b kn m

explicit characterization of the drazin index abstract let b x be the set of bounded linear operators on a banach space x and a b x be drazin invertible an element b b x is said to be a stable perturbation of a if b is drazin invertible and i a b is invertible where i is the identity operator on x a and b are the spectral projectors of a and b respectively under the condition that b is a stable perturbation of a a formula for the drazin inverse b d is derived based on this formula a new approach is provided to the computation of the explicit drazin indices of certain 2 2 operator matrices

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

summing the spectral representations of poschl teller and rosen morse fixed energy amplitudes the spectral representations of the fixed energy amplitude of the symmetric and the general poschl teller potentials are summed via a sommerfeld watson transformation which leads to a simple closed form expression the result is used to write down a similar expression for the symmetric and general rosen morse potentials exploiting the close correspondence that exists between the two systems within the schrodinger theory and the path integral formalism via a duru kleinert transformation from the singularities of the latter amplitude the bound and continuum states of the general rosen morse potential are extracted

on the distribution of estimators of diffusion constants for brownian motion we discuss the distribution of various estimators for extracting the diffusion constant of single brownian trajectories obtained by fitting the squared displacement of the trajectory the analysis of the problem can be framed in terms of quadratic functionals of brownian motion that correspond to the euclidean path integral for simple harmonic oscillators with time dependent frequencies explicit analytical results are given for the distribution of the diffusion constant estimator in a number of cases and our results are confirmed by numerical simulations

free boundary problems and the set valued mappings we have pointed out that many free boundary problems may be reduced to find the fixed points of certain set valued mappings and have solved the free boundary problem arising in the plasma physics 7 and the free boundary problem of the water cone in an oil reservoir with bottom water 111 as examples it is the purpose of this paper to develop the idea more extensively in this paper some typical free boundary problems such as the obstacle problem the seepage surface problem the stefan problem the elenbaas equation etc are all picked up into our scheme although most of them have been attacked by other methods there are still some new things in our approach especially the multiplicity of the solutions of the elenbaas equation is interesting we prove that there are l 1 and two distinct positive solutions of the equation

characterization of ray pattern matrix whose determinantal region has two components after deleting the origin abstract ray nonsingular matrices are generalizations of sign nonsingular matrices the problem of characterizing ray nonsingular matrices is still open the study of the determinantal regions r a of ray pattern matrices is closely related to the study of ray nonsingular matrices it was proved that if r a 0 is disconnected then it is a union of two opposite open sectors or open rays in this paper we characterize those ray patterns whose determinantal regions become disconnected after deleting the origin the characterization is based on three classes f1 f2 and f3 of matrices which can further be characterized in terms of the sets of the distinct signed transversal products of their ray patterns moreover we show that in the fully indecomposable case a matrix a is in the class f1 or f2 respectively if and only if a is ray permutation equivalent to a real sns or non sns respectively matrix

several expressions for the generalized drazin inverse of a block matrix in a banach algebra we present some new representations for the generalized drazin inverse of a block matrix with generalized schur complement being generalized drazin invertible in a banach algebra under conditions weaker than those used in recent papers on the subject

heat equations with discontinuous nonlinearities on convex and nonconvex constraints in this paper we use a variational method for solving some semilinear parabolic equations with a discontinuous nonlinearity possibly on either some convex or nonconvex constraints the approach is based on the fact that the solutions of the above mentioned problems can be viewed as steepest descent curves in a suitable sense to be specified for some lower semicontinuous functionals possibly restricted to suitable constraints this abstract framework which is presented in section 1 seems interesting to us in that it provides a unifying tool for treating various kinds of constrained problems including cases in which the constraint is not convex see section 4 moreover the existence theorem that we get holds under reasonably weak assumptions all these ideas in great part originated from the paper lo where a general framework for variational evolution was proposed and are also related to the theory of maximal monotone operators see 4 and some of its extensions see 5 6 9 11 15 17 19 241 for some applications the main difference being in the fact that using compactness we find existence theorems without uniqueness the applications presented can be described roughly speaking as follows given an open set sz c rn and g q x r r possibly discontinuous we search for u 0 t t2 sz which solve wf e g q u t a t v t in i and a e in i

ray patterns of matrices and nonsingularity a complex matrix a is ray nonsingular if det x 0 a f 0 for every matrix x with positive entries a sufficient condition for ray nonsingularity is that the origin is not in the relative interior of the convex hull of the signed transversal products of a the concept of an isolated set of transversals is defined and used to obtain a neces

g outer inverse of banach spaces operators abstract the aim of this paper is to present a new generalized inverse which is called a g outer inverse of operators between two banach spaces as a generalization of g drazin inverses of square matrices and banach space operators several characterizations of this new inverse are given based on the g outer inverse we define and characterize the g outer partial order extending the g drazin partial order to operators between two banach spaces we also relate this new partial order to the minus partial order in the end we study an outer binary relation which is an equivalence relation on the corresponding set

a note on the representation for the drazin inverse of 2 2 block matrices abstract in this note we consider representations of the drazin inverse of 2 2 block matrices under conditions weaker than those used in recent papers on the subject in particular in d s djordjevic p s stanimirovic on the generalized drazin inverse and generalized resolvent czechoslovak math j 51 126 2001 617 634 r hartwig x li y wei representations for the drazin inverse of 2 2 block matrix siam j matrix anal appl 27 2006 757 771

rational realizations of the minimum rank of a sign pattern matrix abstract a sign pattern matrix is a matrix whose entries are from the set 0 the minimum rank of a sign pattern matrix a is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of a it is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix the equivalence of this conjecture to several seemingly unrelated statements are established for some special cases such as when a is entrywise nonzero or the minimum rank of a is at most 2 or the minimum rank of a is at least n 1 where a is m n the conjecture is shown to hold connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either 1 or 1 are investigated

matrices with signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a in this paper we give complete characterizations of the m n matrices a which have signed generalized inverses for both cases a n and a where a is the term rank of a and without loss of generality we assume n m and thus solve a problem proposed in b l shader siam j matrix anal appl 16 1995 1056 using these characterizations we are also able to show that the property of having a signed generalized inverse for a matrix a is inherited by all the submatrices b of a with b a and is also inherited by all those matrices a 1 with a 1 a which can be obtained from a by replacing some nonzero entries of a by zero we also consider several special cases of a problem proposed in r a brualdi b l shader matrices of sign solvable linear systems cambridge university press cambridge ma 1995 b l shader siam j matrix anal appl 16 1995 1056 about the characterization of the matrices in a special triangular block form to have signed generalized inverses

graphical description of group inverses of certain bipartite matrices abstract a necessary and sufficient condition for the existence of the group inverse of a certain bipartite matrix is given and an explicit formula is obtained for the group inverse in terms of its block submatrices this form is used to derive a graph theoretic description of the entries of the group inverse of some examples of such a matrix including those corresponding to broom graphs if the group inverse of a nonnegative matrix corresponding to a broom graph exists then it is shown that this group inverse is signed an open question about group inverses of more general bipartite matrices is posed and a summary of cases for which its answer is known is given

the drazin inverse of an even order tensor and its application to singular tensor equations abstract the notion of the moore penrose inverses of matrices was recently extended from matrix space to even order tensor space with einstein product in the literature in this paper we further study the properties of even order tensors with einstein product we define the index and characterize the invertibility of an even order square tensor we also extend the notion of the drazin inverse of a square matrix to an even order square tensor an expression for the drazin inverse through the core nilpotent decomposition for a tensor of even order is obtained as an application the drazin inverse solution of the singular linear tensor equation a x b will also be included

some additive results on drazin inverses in this paper some additive results on drazin inverse of a sum of drazin invertible elements are derived some converse results are also presented

a note on the representations for the drazin inverse of 2 2 block matrices abstract in 1979 campbell and meyer proposed the problem of finding a formula for the drazin inverse of a 2 2 matrix m a b c d in terms of its various blocks where the blocks a and d are required to be square matrices special cases of the problems have been studied in particular a representation of the drazin inverse of m denoted by md has recently been obtained under the assumptions that c i aad b o and a i aad b o together with the condition that the generalized schur complement d cadb be either nonsingular or zero we derive an alternative representation for md under the same assumptions but with the condition on the schur complement in the hypothesis replaced by the condition that r caa d n b n d where r and n are the range and null space of a matrix

minimum maximum rank of sign pattern tensors and sign nonsingular tensors abstract in this paper we define the sign pattern tenors minimum maximum rank of sign pattern tenors term rank of tensors and sign nonsingular tensors the necessity and sufficiency for the minimum rank of sign pattern tenors to be 1 is given we show that the maximum rank of a sign pattern tensor is not less than the term rank and the minimum rank of the sign pattern of a sign nonsingular tensor is not less than its dimension we get some characterizations of tensors having sign left or sign right inverses

complex neural network models for time varying drazin inverse two complex zhang neural network znn models for computing the drazin inverse of arbitrary time varying complex square matrix are presented the design of these neural networks is based on corresponding matrix valued error functions arising from the limit representations of the drazin inverse two types of activation functions appropriate for handling complex matrices are exploited to develop each of these networks theoretical results of convergence analysis are presented to show the desirable properties of the proposed complex valued znn models numerical results further demonstrate the effectiveness of the proposed models

representations of the drazin inverse for a class of block matrices abstract in this paper we give a formula for the drazin inverse of a block matrix f i i e 0 where e is square and i is the identity matrix further we get representations of the drazin inverse for matrices of the form m a b c 0 where a is square under some conditions expressed in terms of the individual blocks

a viability theorem for set valued states in a hilbert space abstract applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact possibly non convex sets rather than vectors this leads to evolutions in a basic set which can be supplied with a metric like the well established pompeiu hausdorff distance but it does not have an obvious linear structure this article extends differential inclusions with state constraints to compact valued states in a separable hilbert space h the focus is on sufficient conditions such that a given constraint set of compact subsets is viable a k a weakly invariant our main result extends the tangential criterion in the well known viability theorem usually for differential inclusions in a vector space to the metric space of non empty compact subsets of h

some additive results on drazin inverse some additive perturbation results for drazin inverses are given in particular a formula is given for the drazin inverse of a sum of two matrices when one of the products of these matrices vanishes some special applications of this are also considered

the ray nonsingularity of certain uniformly random ray patterns abstract a uniformly random ray pattern matrix a with a given zero nonzero pattern described by a digraph d with no multi arcs or loops is the matrix whose nonzero entries are mutually independent random variables uniformly distributed over the unit circle s 1 in the complex plane it is shown in this paper that the probability of i a to be ray nonsingular is completely determined by the cycle graph cg d of d i e the adjacency structure of the directed cycles in d if cg d is a tree a formula is given to compute the probability when cg d is a tree and it is also shown that as the order of cg d tends to infinity the limit of the probability is 0

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

impulse capture basins of sets under impulse control systems given a target contained in a constrained set and an impulse control system governing the evolutions of runs or executions that are hybrids of continuous and discrete evolutions this paper studies and provides several characterizations of the capture basin of the target viable in the constrained set it is the subset of initial runs from which start at least one run viable in the constrained set until it reaches the target in finite time it also provides algorithms and regulation rules governing the runs that reach the targets while obeying state constraints

condition number for the drazin inverse and the drazin inverse solution of singular linear system with their condition numbers in this paper we investigate the condition number of drazin inverse and drazin inverse solution of singular linear system ax b where a is a nxn rank deficient matrix and b a real vector of size n x a real vector let a and b be two positive real numbers when we consider the weighted frobenius norm aa bb p q f on the data we get the formula of condition number of the drazin inverse solution of singular linear system for the normwise condition number the sensitivity of the relative condition number itself is studied the componentwise perturbation is also investigated

on the existence of group inverses of peirce corner matrices abstract we give some statements that are equivalent to the existence of group inverses of peirce corner matrices of a 2 2 block matrix and its generalized schur complements as applications several new results for the drazin inverses of the generalized schur complements and the 2 2 block matrix are obtained and some of them generalize several results in the literature

relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices application of perturbation theory for simple invariant subspaces perturbation bounds for the relative error in the eigenvalues of diagonalizable and singular matrices are derived by using perturbation theory for simple invariant subspaces of a matrix and the group inverse of a matrix these upper bounds are supplements to the related perturbation bounds for the eigenvalues of diagonalizable and nonsingular matrices

ray pattern of drazin inverses of some matrices abstract let q a be the set of complex matrices with the same ray pattern as a for any a q a if the drazin inverses of a and a have the same ray pattern then a is said to have ray drazin inverse in this paper we give some properties on matrices whose drazin inverses or m p inverses are ray unique and use these properties to obtain necessary and sufficient conditions for some block matrices to have ray drazin inverses

on some sets of permutation matrices abstract in 5 a class of p p circulant matrices was studied where p is a prime and necessary and sufficient conditions were presented for the matrices in to be commutative and to be closed with respect to matrix multiplication here we show that these properties also hold for n n circulant matrices where n is a positive integer with an additional condition namely contains an n cycle

on the drazin inverse of block matrices and generalized schur complement in this paper we give an additive result for the drazin inverse with its applications we obtain representations for the drazin inverse of a 2x2 complex block matrix having generalized schur complement s d ca db equal to zero or nonsingular several situations are analyzed and recent results are generalized r e hartwig x li y wei representations for the drazin inverse of a 2x2 block matrix siam j matrix anal appl 27 3 2006 757 771

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

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

ray pattern matrices requiring perron frobenius properties abstract a ray pattern matrix is a matrix whose nonzero entries are all unimodular an n n ray pattern matrix a is said to require the weak perron frobenius property if for any n n entrywise positive matrix k the spectral radius k a is one of the eigenvalues of k a to require the strong perron frobenius property if k a is an algebraically simple eigenvalue of k a and the left and right eigenvectors associated to k a are strictly nonzero in this paper necessary and sufficient conditions for ray pattern matrices requiring the weak strong perron frobenius property are given

the moore penrose inverse of matrices with an acyclic bipartite graph abstract the moore penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs for such a matrix the sign of every entry of the moore penrose inverse is shown to be determined uniquely by the signs of the matrix entries i e the matrix has a signed generalized inverse necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative moore penrose inverse nearly reducible matrices are proved to contain no submatrix having two or more diagonals implying that a nearly reducible matrix has a signed generalized inverse furthermore it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix

an analytic treatment of the first order correction to the poisson boltzmann interaction free energy in the case of counterion only coulomb fluid the author has computed the first order correlation correction to the poisson boltzmann result for the interaction energy of two charged surfaces immersed into a counterion only coulomb fluid the cases of small and large coupling were amenable to analytic treatment in the saddle point approximation

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

on additive properties of the drazin inverse of block matrices and representations abstract in this paper we give a new additive formula for the drazin inverse under conditions weaker than those used in some current literature on this subject also we obtain representations for the drazin inverse of a complex block matrix having generalized schur complement equal to zero

from boolean to sign pattern matrices abstract a nonnegative sign pattern matrix is a matrix whose entries are from the set 0 a nonnegative sign pattern matrix can also be viewed as a boolean matrix by replacing each entry with 1 in this paper some interesting connections between nonnegative sign pattern matrices and boolean matrices are investigated in particular the relations between the minimum rank and the boolean row or column rank are explored the idempotent boolean matrices that allow idempotence are identified and the nonnegative sign patterns that allow various types of nonnegative or positive generalized inverses are characterized

on sign inconsistent linear systems abstract a linear system of equations ax b is called sign inconsistent if for each matrix a b with the same sign pattern as a b the corresponding linear system a x b is not solvable sign inconsistent linear systems have close relationships with sign solvable and conditionally sign solvable linear systems in this paper we study various properties of sign inconsistent linear systems we give a complete characterization of sign inconsistent linear systems in terms of l matrices we also obtain complete characterizations of the minimally sign inconsistent linear systems in terms of the l canonical forms and their corresponding canonical digraphs of the barely l matrices

the minimum rank problem over the finite field of order 2 minimum rank 3 our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2 we also list all 62 such minimal forbidden subgraphs we conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

idempotence for sign pattern matrices abstract we first characterize the n by n irreducible sign pattern matrices a that are sign idempotent that is a a 2 to identify the n by n reducible sign idempotent patterns we develop the upper diagonal completion process to find the sign pattern of each off diagonal block in an upper block triangular sign pattern matrix a so that a a 2 next we analyze the completion process qualitatively and then discuss a graph theoretic interpretation of it we then formulate the first characterization of sign idempotent patterns in terms of the upper diagonal completion process finally we establish a graph theoretic characterization of sign idempotent patterns

sign pattern matrices and generalized inverses abstract if a is an n n sign pattern matrix then q a denotes the set of all real n n matrices b such that the sign of the entries in b match the corresponding entries in a in this paper we consider various requires allows questions connected with sign pattern matrices and generalized inverses in particular we investigate the class of all square patterns a for which there exist b c q a where bcb b for nonnegative patterns we characterize and show that coincides with the class of all square patterns a for which there exists b q a where b3 b nonnegative square patterns that allow an idempotent and those that allow a 1 3 inverse are each characterized some interesting open questions are also indicated

the drazin inverse of bounded operators with commutativity up to a factor abstract we explore the drazin inverse of bounded operators with commutativity up to a factor pq qp in a banach space conditions on drazin invertibility are formulated and shown to depend on spectral properties of the operators involved we also present a result concerning the more general problem of commutativity up to a related operator factor pq pqp under the condition of commutativity up to a factor pq qp resp pq pqp we give explicit representations of the drazin inverse p q d resp p q d in term of p p d q and q d

representations for the drazin inverse of 2 2 block operator matrix with singular schur complement abstract in this paper we present formulae for the drazin inverse of 2 2 block operator matrix with generalized schur complement being drazin invertible moreover necessary and sufficient conditions for the existence as well as the expressions for the group inverse are obtained

a note on the drazin inverses with banachiewicz schur forms let h be a hilbert space m the closed subspace of h with orthocomplement m according to the orthogonal decomposition email protected m every operator email protected b h can be written in a block form m abcd in this note we give necessary and sufficient conditions for a partitioned operator matrix m to have the drazin inverse with banachiewicz schur form in addition this paper investigates the relations among the drazin inverse the moore penrose inverse and the group inverse when they can be expressed in the banachiewicz schur forms

acute perturbation of the group inverse abstract for any n n complex matrix a let a g be the group inverse of a when a is singular a matrix b a e is said to be an acute perturbation of a if e is small and the spectral radius b g b a g a 1 the acute perturbation coincides with the stable perturbation of the group inverse if the matrix b satisfies condition r b n a 0 n b r a 0 which was introduced by castro gonzalez et al 2008 8 furthermore several examples are provided to illustrate the acute perturbation of the group inverse

nagumo type condition for partial differential inclusions on etudie un probleme de la forme x t ax t g xc h x o x 0 k t o t x t k ou a est un operateur elliptique et k n est pas necessairement compact on trouve des conditions tangentielles de type nagumo

the drazin inverse in multibody system dynamics the numerical analysis of multibody system dynamics is based on the equations of motion as differential algebraic systems a thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches the main tool is the drazin inverse a generalized matrix inverse which preserves the eigenvalues the results are illustrated by a realistic truck model finally the approach is extended to the nonlinear index 2 formulation

formulas for the drazin inverse of special block matrices properties of the drazin inverse of the matrix f iie0 with e square are investigated based on this approach it is obtained an explicit formula for the drazin inverse of matrices of the form a ip tquv t where u v p and q are nxk the representation for a d is given in terms of kxk matrices involving the individual blocks under some conditions some special cases are also analyzed

minimum ranks of sign patterns and zero nonzero patterns and point hyperplane configurations abstract a sign pattern matrix is a matrix whose entries are from the set 0 the minimum rank respectively rational minimum rank of a sign pattern matrix a is the minimum of the ranks of the real respectively rational matrices whose entries have signs equal to the corresponding entries of a a sign pattern a is said to be condensed if a has no zero row or column and no two rows or columns are identical or negatives of each other a zero nonzero pattern matrix is a matrix whose entries are from the set 0 where indicates a nonzero entry many of the sign pattern notions carry over to zero nonzero patterns assuming that the ground field is r in this paper a direct connection between condensed m n sign patterns and zero nonzero patterns with minimum rank r and m point n hyperplane configurations in r r 1 is established in particular condensed sign patterns and zero nonzero patterns with minimum rank 3 are closely related to point line configurations on the plane using this connection we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3 it is proved that for any sign pattern or zero nonzero pattern a if the number of zero entries on each column of a is at most 2 then the rational and real minimum ranks of a are equal further it is shown that for any zero nonzero pattern a with minimum rank r 3 if the number of zero entries on each column of a is at most r 1 then the rational minimum rank of a is also r a few related conjectures and open problems are raised

additive perturbation results for the drazin inverse abstract in this paper some new additive results for the drazin inverse are presented we give a formula for the drazin inverse of a sum of two matrices under conditions on the matrices less restrictive than those imposed in the corresponding theorem given by hartwig et al linear algebra appl 322 2001 207 217 we consider some aplications of our results to the perturbation of the drazin inverse and analyze a number of special cases

viability theorems for a class of differential operator inclusions abstract this paper discusses the following viability problem of a differential inclusion x t ax t f x t x 0 x0 k t 0 x t k where k is a compact subset of a banach space x a is the infinitesimal generator of a compact differentiable semigroup of bounded linear operators on x and f k x is upper semicontinuous with compact convex values we show that a natural tanegential condition is necessary and sufficient for the existence of a global solution to this problem

an expression of the drazin inverse of a perturbed matrix given a square matrix a and its perturbation matrix e a new expression for the drazin inverse b d of b a e is derived if aa db 2 aa db 2 or b 2aa d baa d 2 based on the new expression a bound of the relative error of b d is developed some known results in the literature on the drazin inverse and the perturbation bound are included by this new formula as special cases a numerical example is given to compare the upper bounds

matrices with doubly signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a a is said to have doubly signed generalized inverse if both a and a have a signed generalized inverse matrices with doubly signed generalized inverses are generalizations of those matrices a such that both a and a 1 are s 2 ns matrices which are studied in matrices of sign solvable linear systems cambridge university press cambridge 1995 linear algebra appl 232 1996 97 in this paper we give a complete characterization of matrices with doubly signed generalized inverses we show that if a is a real matrix with no zero rows and no zero columns then a has doubly signed generalized inverses if and only if a is permutation equivalent to a direct sum of matrices of the following types a row or a column with no zero entries fully indecomposable s 2 ns matrices of order 1 or 2 matrices of the blocked form d 1 o x d 2 where d 1 and d 2 are both invertible diagonal matrices

expressions for the g drazin inverse of additive perturbed elements in a banach algebra abstract we study additive properties of the g drazin inverse in a banach algebra a in our development we derive a representation of the resolvent of a 2 2 matrix with entries in a which is then used to find explicit expressions for the g drazin inverse of the sum a b under new conditions on a b a as an application of our results we obtain a representation for the drazin inverse of a 2 2 complex block matrix in terms of the individual blocks under certain conditions

representations for the drazin inverses of 2 2 block matrices abstract let m denote a 2 2 block complex matrix a b c d where a and d are square matrices not necessarily with the same orders in this paper explicit representations for the drazin inverse of m are presented under the condition that bd i c 0 for i 0 1 n 1 where n is the order of d

a note on the drazin inverse of an anti triangular matrix abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m under some conditions moreover some particular cases of these results are also considered

on the period and base of a sign pattern matrix abstract a square sign pattern matrix a whose entries are or 0 is said to be powerful if all the powers a 1 a 2 a 3 are unambiguously defined for a powerful pattern a if a l a l p with l and p minimal then l is called the base of a and p is called the period of a we characterize irreducible powerful sign pattern matrices and investigate the period and base of a powerful sign pattern matrix we also consider some connections with real matrices and give some significant classes of powerful patterns