the inverse eigenvalue problem of generalized reflexive matrices and its approximation national natural science foundation of china 10961010 natural science foundation of jiangxi china 2007gzs1760 jiangxi education office china gjj08432

on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

the constrained inverse eigenvalue problem and its approximation for normal matrices abstract in this paper the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered the solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented and the expression of the solution for the optimal approximation problem is obtained

on the bridge between combinatorial optimization and nonlinear optimization a family of semidefinite bounds for 0 1 quadratic problems leading to quasi newton methods this article presents a family of semidefinite programming bounds obtained by lagrangian duality for 0 1 quadratic optimization problems with linear or quadratic constraints these bounds have useful computational properties they have a good ratio of tightness to computing time they can be optimized by a quasi newton method and their final tightness level is controlled by a real parameter these properties are illustrated on three standard combinatorial optimization problems unconstrained 0 1 quadratic optimization heaviest k subgraph and graph bisection

distributed computation of linear matrix equations an optimization perspective this paper investigates the distributed computation of the well known linear matrix equation in the form of axb f with the matrices a b x and f of appropriate dimensions over multiagent networks from an optimization perspective in this paper we consider the standard distributed matrix information structures where each agent of the considered multiagent network has access to one of the subblock matrices of a b and f to be specific we first propose different decomposition methods to reformulate the matrix equations in standard structures as distributed constrained optimization problems by introducing substitutional variables we show that the solutions of the reformulated distributed optimization problems are equivalent to least squares solutions to original matrix equations and we design distributed continuous time algorithms for the constrained optimization problems even by using augmented matrices and a derivative feedback technique moreover we prove the exponential convergence of the algorithms to a least squares solution to the matrix equation for any initial condition

theory and method for updating least squares finite element model of symmetric generalized centro symmetric matrices let x f be a displacement matrix and load matrix respectively c obtained by calculations or measurements is an estimate matrix of the analytical model a method is presented for correction of the model c based on the theory of inverse problem of matrices the corrected model is symmetric generalized centro symmetric with specified displacements and loads satisfying the mechanics characters of finite element model the application of the method is illustrated it is more important that a perturbation analysis is given which is not given in the earlier papers numerical results show that the method is feasible and effective

least squares solution of f pg over positive semidefinite symmetric p abstract a matrix optimization problem of interest is the infimization for arbitrary f g rn m of f pg with respect to positive semidefinite symmetric p rn n where denotes the frobenius norm one motivation for its study is that solutions may be used as estimates of the inverse hessian of a nonlinear differentiable function f rn r1 which is to be minimized with respect to a parameter vector x rn by a quasi newton type algorithm another is where the compliance matrix of an elastic structure is to be estimated from experimental measurements of the displacements resulting from some static loading necessary and sufficient conditions for the existence of the minimum are given a sufficient condition is that rank g n which is also necessary and sufficient for uniqueness of minimizers arbitrarily accurate infimization can be achieved via solution of an instance of the minimization problem where rank g n however no general solution procedure for the latter is known which motivates algorithmic solution an algorithm is given which for the case rank g n arbitrarily accurately approximates the unique minimizer the algorithm is globally convergent requiring the computation of inner products and solutions of symmetric eigenvalue problems

an iterative method for the symmetric and skew symmetric solutions of a linear matrix equation axb cyd e in this paper two efficient iterative methods are presented to solve the symmetric and skew symmetric solutions of a linear matrix equation axb cyd e respectively with real pair matrices x and y by these two iterative methods the solvability of the symmetric and skew symmetric solutions for the matrix equation can be determined automatically when the matrix equation has symmetric and skew symmetric solutions then for any initial pair matrices x 0 and y 0 symmetric and skew symmetric solutions can be obtained within finite iteration steps in the absence of roundoff errors and the minimum norm of the symmetric and skew symmetric solutions can be obtained by choosing a special kind of initial pair matrices in addition the unique optimal approximation pair solution x and y to the given matrices x and y in frobenius norm can be obtained by finding the minimum norm solution of a new matrix equation ax b cy d e where e e ax b cy d the given numerical examples demonstrate that the iterative methods are quite efficient

the inverse eigenproblem with a submatrix constraint and the associated approximation problem for r s symmetric matrices abstract let r r n n and s r n n be nontrivial involutions i e r r 1 i and s s 1 i a matrix a r n n is called r s symmetric if r a s a this paper presents a r s symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint the solvability condition of the constrained inverse eigenproblem is also derived the existence the uniqueness and the expression of the r s symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved respectively an algorithm is presented to compute the r s symmetric matrix solution to the best approximation problem two numerical examples are given to illustrate the effectiveness of our results

successive projection iterative method for solving matrix equation ax b in this paper we present a new iterative method successive projection iterative method to solve matrix equation ax b where a is a symmetric positive definite spd matrix based on this method an algorithm is proposed and proved to be convergent in addition analysis of the algorithm and numerical experiments are also given to show the efficiency of the method

inverse eigenvalue problems of tridiagonal symmetric matrices and tridiagonal bisymmetric matrices the problem of generating a matrix a with specified eigenpairs where a is a tridiagonal symmetric matrix is presented a general expression of such a matrix is provided and the set of such matrices is denoted by s e moreover the corresponding least squares problem under spectral constraint is considered when the set s e is empty and the corresponding solution set is denoted by s l the best approximation problem associated with s e s l is discussed that is to find the nearest matrix a in s e s l to a given matrix the existence and uniqueness of the best approximation are proved and the expression of this nearest matrix is provided at the same time we also discuss similar problems when a is a tridiagonal bisymmetric matrix

iterative algorithm for solving a class of convex feasibility problem abstract in this paper we consider a class of convex feasibility problem which arises in quantum computation based on the matrix equation theory the feasible sets are characterized by exploiting the special structure of the linear constraints and its analytic expression is given by making use of the nice structure properties and the kkt condition we derive the projection formulas of a matrix onto the feasible sets the relaxed alternating projection method is designed to solve the convex feasibility problem numerical experiments show that the new method is feasible and effective

procrustes problems and associated approximation problems for matrices with k involutory symmetries abstract we say that a matrix r c n n is k involutary if its minimal polynomial is x k 1 for some k 2 so r k 1 r 1 and the eigenvalues of r are 1 2 k 1 where e 2 i k let 0 1 k 1 if r c m m a c m n s c n n and r and s are k involutory we say that a is r s symmetric if ras 1 a and a is r s symmetric if ras a let l be the class of m n r s symmetric matrices or the class of m n r s symmetric matrices given x c n t and b c m t we characterize the matrices a in l that minimize ax b frobenius norm and given an arbitrary w c m n we find the unique matrix a l that minimizes both ax b and a w we also obtain necessary and sufficient conditions for existence of a l such that ax b and assuming that the conditions are satisfied characterize the set of all such a

the left and right inverse eigenvalue problems of generalized reflexive and anti reflexive matrices let nxn complex matrices r and s be nontrivial generalized reflection matrices i e r r r 1 i n s s s 1 i n a complex matrix a with order n is said to be a generalized reflexive or anti reflexive matrix if ras a or ras a in this paper the solvability conditions of the left and right inverse eigenvalue problems for generalized reflexive and anti reflexive matrices are derived and the general solutions are also given in addition the associated approximation solutions in the solution sets of the above problems are provided the results in present paper extend some recent conclusions

least square solutions for inverse problems of centrosymmetric matrices let csrnxn a aij rnxn aij an 1 j i j 1 2 n in this paper we mainly discuss solving the following problems problem io given x b rnxm find a csrn n such that ax b problem i given x b rn m find a csrn n such that ax b min problem ii given a rn n find a se such that where is the frobenius norm and se is the solution set of problem i or i0 the expressions of the general solution of problems i and 10 have been derived the sufficient and necessary conditions for the solvability of problem i0 have been provided the numerical algorithm to find the optimal approximate solution and some numerical experiments have been given

an iterative method for the least squares symmetric solution of the linear matrix equation axb c abstract this paper an iterative method is presented to solve the minimum frobenius norm residual problem min axb c with unknown symmetric matrix x by this iterative method for any initial symmetric matrix x0 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial symmetric matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by first finding the least norm solution x of the new minimum residual problem min a x b c with unknown symmetric matrix x where c c a x x t 2 b given numerical examples are show that the iterative method is quite efficient

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

realizations of frobenius functions abstract in an earlier paper agoston et al cms conf proc 18 1996 17 37 we studied the so called frobenius functions on certain translation quivers here we show that the classification given there is in some sense complete every frobenius length function on the wing w n and the tube t n is equivalent to the length function on a convex subquiver of the auslander reiten quiver of the module category over some algebra a

positive semidefinite matrices characterization via conical hulls and least squares solution of a matrix equation any real symmetric n times n matrix a can be described by an n n 1 2 component vector positive semidefiniteness of a is characterized by the associated vector belonging to the conical hull of a suitable convex set this characterization is used to facilitate least squared error solution with respect to such a of f ag where f and g are given matrices the solution method involves finding the point in the conical hull of a convex set which is nearest to a vector an algorithm is given for solving that proximal point problem

on the cone of positive semidefinite matrices abstract a survey of some general properties of the cone of positive semidefinite matrices its faces two isometric isomorphisms and linear transformations on it is given

left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem in this paper the left and right inverse eigenpairs problem for generalized centrosymmetric matrices is considered we obtain the necessary and sufficient conditions for the solvability of the problem and we present the general expression of the solution the related optimal approximation problem to a given matrix over the solution set is solved in addition a numerical algorithm and examples to solve the problem are given

approximating the nearest stable discrete time system abstract in this paper we consider the problem of stabilizing discrete time linear systems by computing a nearby stable matrix to an unstable one to do so we provide a new characterization for the set of stable matrices we show that a matrix a is stable if and only if it can be written as a s 1 u b s where s is positive definite u is orthogonal and b is a positive semidefinite contraction that is the singular values of b are less or equal to 1 this characterization results in an equivalent non convex optimization problem with a feasible set on which it is easy to project we propose a very efficient fast projected gradient method to tackle the problem in variables s u b and generate locally optimal solutions we show the effectiveness of the proposed method compared to other approaches

the solvability conditions for the inverse eigenproblems of symmetric and generalized centro symmetric matrices and their approximations let be the frobenius norm of matrices we consider i the set se of symmetric and generalized centro symmetric real n n matrices rn with some given eigenpairs j qj j 1 2 m and ii the element r in se which minimizes r r for a given real matrix r necessary and sufficient conditions for se to be nonempty are presented a general form of elements in se is given and an explicit expression of the minimizer r is derived finally a numerical example is reported

matrix inverse problem and its optimal approximation problem for r symmetric matrices abstract let r cn n be a nontrivial involution i e r2 i and r i a matrix a cn n is called r skew symmetric if rar a the least squares solutions of the matrix inverse problem for r skew symmetric matrices with r r are firstly derived then the solvability conditions and the solutions of the matrix inverse problem for r skew symmetric matrices with r r are given the solutions of the corresponding optimal approximation problem with r r for r skew symmetric matrices are also derived at last an algorithm for the optimal approximation problem is given it can be seen that we extend our previous results g x huang f yin matrix inverse problem and its optimal approximation problem for r symmetric matrices appl math comput 189 2007 482 489 and the results proposed by zhou et al f z zhou l zhang x y hu least square solutions for inverse problem of centrosymmetric matrices comput math appl 45 2003 1581 1589

left and right inverse eigenvalue problem of r s symmetric matrices and its optimal approximation problem the left and right inverse eigenvalue problem which mainly arises in perturbation analysis of matrix eigenvalue and recursive matters has some practical applications in engineer and scientific computation fields in this paper we give the solvability conditions of and the general expressions to the left and right inverse eigenvalue problem for the r s symmetric and r s skew symmetric solutions the corresponding best approximation problems for the left and right inverse eigenvalue problem are also solved that is given an arbitrary complex n by n matrix a find a r s symmetric or r s skew symmetric matrix a a which is the solution to the left and right inverse eigenvalue problem such that the distance between a and a a is minimized in the frobenius norm we give an explicit solution to the best approximation problem in the r s symmetric and r s skew symmetric solution sets of the left and right inverse eigenvalue problem under the assumption that r r and s s a numerical example is given to illustrate the effectiveness of our method

an inverse eigenproblem and an associated approximation problem for generalized reflexive and anti reflexive matrices in this paper we first give the existence of and the general expression for the solution to an inverse eigenproblem defined as follows given a set of real n vectors x i i 1 m and a set of real numbers l i i 1 m and an n by n real generalized reflexive matrix a or generalized anti reflexive matrix b such that x i i 1 m and l i i 1 m are the eigenvectors and eigenvalues of a or b respectively we solve the best approximation problem for the inverse eigenproblem that is given an arbitrary real n by n matrix email protected we find a matrix a a which is the solution to the inverse eigenproblem such that the distance between email protected and a a is minimized in the frobenius norm we give an explicit solution and a numerical algorithm for the best approximation problem over generalized reflexive or generalized anti reflexive matrices two numerical examples are also presented to show that our method is effective

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

the inverse eigenvalue problem for hermitian anti reflexive matrices and its approximation in this paper we first consider the inverse eigenvalue problem as follows find a matrix a with specified eigenpairs where a is a hermitian anti reflexive matrix with respect to a given generalized reflection matrix j the sufficient and necessary conditions are obtained and a general representation of such a matrix is presented we denote the set of such matrices by s a then the best approximation problem for the inverse eigenproblem is discussed that is given an arbitrary a find a matrix email protected s a which is nearest to a in the frobenius norm we show that the best approximation is unique and provide an expression for this nearest matrix

an iteration method to solve multiple constrained least squares problems abstract in this paper we propose an iteration method to solve the multiple constrained least squares matrix problem we first transform the multiple constrained least squares matrix problems into the multiple constrained matrix optimal approximation problem and then we use the idea of dykstra s algorithm to derive the basic iterative pattern we observe that we only need to solve multiple single constrained least squares matrix problems at each iteration step of the proposed algorithm we give a numerical example to illustrate the effectiveness of the proposed method to solve the original problems also we give an example to illustrate that the method proposed by escalante and li to solve the single constrained least squares matrix problem is not correct

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

tame biserial algebras a similar result is suspected if we drop the property representationfinite in classes 1 and 2 and we will start an investigation of biserial algebras of infinite representation type the key to the result for representation finite algebras was to observe that each representation finite biserial algebra is special 13 see also the basic notations below which means that it can be presented by a quiver with nice relations special algebras play an important role in the modular representation theory of finite groups namely each representation finite block of a group algebra and some of the tame blocks they occur only in characteristic 2 are special 7 10 4 moreover in the complex representation theory of the lorentz group the so called harish chandra modules are defined over tame special algebras 6 in this paper we show that l a 62 for any special algebra a and that special algebras of infinite representation type are tame our paper consists of four parts first we prove that each special algebra is a factor of a special symmetric algebra theorem 1 5 for the latter the methods of gelfand and ponomarev used in the classification of the indecomposable harish chandra modules of the lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras proposition 2 3 presented in section 2 from this list we 480 0021 8693 85 3 00

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

the solvability conditions for the inverse eigenvalue problems of centro symmetric matrices this paper involves related inverse eigenvalue problems of centro symmetric matrices and their optimal approximation the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue and the general provided form of the solution furthermore the algorithm to compute the optimal approximate solution and some numerical experiments are given

matrix equations over r s symmetric and r s skew symmetric matrices let email protected c m x m and email protected c n x n be nontrivial involution matrices i e r r 1 i and s s 1 i an mxn complex matrix a is said to be a r s symmetric r s skew symmetric matrix if ras a ras a the r s symmetric and r s skew symmetric matrices have many special properties and are widely used in engineering and scientific computations in this paper we consider the matrix equations a 1xb 1 c a 1x d 1 xb 2 d 2 and a 1x d 1 xb 2 d 2 a 3x d 3 xb 4 d 4 over the r s symmetric r s skew symmetric matrix x we derive necessary and sufficient conditions for the existence of r s symmetric r s skew symmetric solutions for these matrix equations also we give the expressions for the r s symmetric r s skew symmetric solutions to the matrix equations

constrained inverse eigenproblem and associated approximation problem for anti hermitian r symmetric matrices abstract let r c n n be a nontrivial involution i e r 2 i and r i a c n n is called anti hermitian r symmetric if a a and rar a the presentation and some properties for an arbitrary anti hermitian r symmetric matrix with r r and the relations between the eigenproblem for a and the corresponding eigenproblems for anti hermitian matrices are given then the solutions of constrained inverse eigenproblem and approximation problem are essentially decomposed into the same kind subproblems for anti hermitian matrices in complex field with smaller dimensions the explicit solutions for the later subproblems are arrived the corresponding problems which are the formulations of constrained inverse eigenproblem and approximation problem in complex field was first given then the solutions of constrained inverse eigenproblem and approximation problem with r r are derived

least squares solution for inverse eigenpair problem of nonnegative definite matrices suppose we know some eigenvalues i and eigenvectors xi associated with i i 1 2 m for a positive semidefinite may be unsymmetric matrix let x x1 x2 xm diag 1 2 m in this paper we mainly discuss solving the following two problems problem i given x rn m diag 1 m find matrices a such that ax x min where a is a positive semidefinite may be unsymmetric matrix problem ii given a rn n find a se such that a a mina se a a where is frobenius norm and se denotes the solution set of problem i an existence theorem of solution for problems i and ii has been given and proved and the general solutions of problem i have been derived sufficient conditions that prove an explicit solution have been provided

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

lyapunov type least squares problems over symmetric cones abstract the lyapunov type least squares problem over symmetric cone is to find the least squares solution of the lyapunov equation with a constraint of symmetric cone in the euclidean jordan algebra and it contains the lyapunov type least squares problem over cone of semidefinite matrices as a special case in this paper we first give a detailed analysis for the image of lyapunov operator in the euclidean jordan algebra relying on these properties together with some characterizations of symmetric cone we then establish some necessary and or sufficient conditions for solution existence of the lyapunov type least squares problem finally we study uniqueness of the least squares solution

the inverse problem for the vibrating beam the beam is modelled by a system of rigid rods joined together by rotational springs and with masses at the joints the left end is clamped while the right end is subject to one of a number of conditions it is free pinned sliding or clamped it is shown that the resonant and antiresonant frequencies of the system may be almost completely ordered necessary and sufficient conditions are found that these frequencies must satisfy to correspond to an actual system with positive parameters and two stripping procedures are devised for the determination of these system parameters from a knowledge of certain frequency spectra

the solvability conditions for the inverse eigenvalue problems of reflexive matrices this paper involves related inverse eigenvalue problems of reflexive matrices and their optimal approximation the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue and the general provided form of the solution furthermore the algorithm to compute the optimal approximate solution and some numerical experiments are given

a new iteration method for the matrix equation ax b an iteration method for the matrix equation ax b is constructed by this iteration method the least norm solution for the matrix equation can be obtained when the matrix equation is consistent and the least norm least squares solutions can be obtained when the matrix equation is not consistent the related optimal approximation solution is also obtained by this iteration method a preconditioned method for improving the iteration rate is put forward finally some numerical examples are given

an inverse eigenvalue problem and an associated approximation problem for generalized k centrohermitian matrices a partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized k centrohermitian matrices are considered it is shown under which conditions the inverse eigenproblem has a solution an expression of its general solution is given in case a solution of the inverse eigenproblem exists the optimal approximation problem can be solved the formula of its unique solution is given

linear parameterized inverse eigenvalue problem of bisymmetric matrices abstract in this paper we consider the linear parameterized inverse eigenvalue problem of bisymmetric matrices which is described as follows problem i given real bisymmetric matrices b 0 b 1 b 2 b n and real numbers 1 2 n find values of c c 1 c 2 c n t such that the eigenvalues of the matrix b c b 0 c 1 b 1 c 2 b 2 c n b n are precisely 1 2 n we first establish the structure of bisymmetric matrices based on the newton type iterative method we then turn problem i into the inverse eigenvalue problems of symmetric matrices with small sizes and solve these problems finally we present numerical examples to illustrate our results

refined error analyses of cholesky factorization this paper concentrates on the practical cholesky process for solving positive definite symmetric systems of linear equations in standard floating point arithmetic two self contained round off analyses are presented which lead to novel a posteriors error bounds of practical use and to refined a priori results of more theoretical significance unmotivated assumptions are avoided with the aim of being constructive throughout

a semi analytical approach for the positive semidefinite procrustes problem abstract the positive semidefinite procrustes psdp problem is the following given rectangular matrices x and b find the symmetric positive semidefinite matrix a that minimizes the frobenius norm of a x b no general procedure is known that gives an exact solution in this paper we present a semi analytical approach to solve the psdp problem first we characterize a family of positive semidefinite matrices that either solve the psdp problem when the infimum is attained or give arbitrary accurate approximations to the infimum when it is not attained this characterization requires the unique optimal solution of a smaller psdp problem where b is square and x is diagonal with positive diagonal elements second we propose a very efficient strategy to solve the psdp problem combining the semi analytical approach a new initialization strategy and the fast gradient method we illustrate the effectiveness of the new approach which is guaranteed to converge linearly compared to state of the art methods

a numerical method of structure preserving model updating problem and its perturbation theory a method is presented to update a special finite element fe analytical model based on matrix approximation theory with spectral constraint at first the model updating problem is treated as a matrix approximation problem dependent on the spectrum data from vibration test and modal parameter identification the optimal approximation is the first modified solution of fe model an algorithm is given to preserve the sparsity of the model by multiple correction the convergence of the algorithm is investigated and perturbation of the modified solution is analyzed finally a numerical example is provided to confirm the convergence of the algorithm and perturbation theory

the quantum mckay correspondence for polyhedral singularities let g be a polyhedral group namely a finite subgroup of so 3 nakamura s g hilbert scheme provides a preferred calabi yau resolution y of the polyhedral singularity 3 g the classical mckay correspondence describes the classical geometry of y in terms of the representation theory of g in this paper we describe the quantum geometry of y in terms of r an ade root system associated to g namely we give an explicit formula for the gromov witten partition function of y as a product over the positive roots of r in terms of counts of bps states gopakumar vafa invariants our result can be stated as a correspondence each positive root of r corresponds to one half of a genus zero bps state as an application we use the crepant resolution conjecture to provide a full prediction for the orbifold gromov witten invariants of 3 g

on solutions of the quaternion matrix equation ax b and their applications in color image restoration by using the complex representation of quaternion matrices and the moore penrose generalized inverse we derive the expressions of the least squares solution with the least norm the least squares pure imaginary solution with the least norm and the least squares real solution with the least norm for the quaternion matrix equation ax b respectively finally we discuss their applications in color image restoration

the crepant transformation conjecture for toric complete intersections abstract let x and y be k equivalent toric deligne mumford stacks related by a single toric wall crossing we prove the crepant transformation conjecture in this case fully equivariantly and in genus zero that is we show that the equivariant quantum connections for x and y become gauge equivalent after analytic continuation in quantum parameters furthermore we identify the gauge transformation involved which can be thought of as a linear symplectomorphism between the givental spaces for x and y with a fourier mukai transformation between the k groups of x and y via an equivariant version of the gamma integral structure on quantum cohomology we prove similar results for toric complete intersections we impose only very weak geometric hypotheses on x and y they can be non compact for example and need not be weak fano or have gorenstein coarse moduli space our main tools are the mirror theorems for toric deligne mumford stacks and toric complete intersections and the mellin barnes method for analytic continuation of hypergeometric functions

hurwitz hodge integrals the e6 and d4 root systems and the crepant resolution conjecture abstract let g be the group a 4 or z 2 z 2 we compute the integral of g on the hurwitz locus h g m g of curves admitting a degree 4 cover of p 1 having monodromy group g we compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the e 6 and d 4 root systems respectively as an application we prove the crepant resolution conjecture for the orbifolds c 3 a 4 and c 3 z 2 z 2

the m n symmetric procrustes problem abstract an p q matrix a is said to be m n symmetric if man man t for given m r n p n r q n in this paper the following m n symmetric procrustes problem is studied find the m n symmetric matrix a which minimizes the frobenius norm of ax b where x and b are given rectangular matrices we use project theorem the singular value decomposition and the generalized singular value decomposition of matrices to analysis the problem and to derive a stable method for its solution the related optimal approximation problem to a given matrix on the solution set is solved furthermore the algorithm to compute the optimal approximate solution and the numerical experiment are given

on the crepant resolution conjecture in the local case in this paper we analyze four examples of birational transformations between local calabi yau 3 folds two crepant resolutions a crepant partial resolution and a flop we study the effect of these transformations on genus zero gromov witten invariants proving the coates iritani tseng ruan form of the crepant resolution conjecture in each case our results suggest that this form of the crepant resolution conjecture may also hold for more general crepant birational transformations they also suggest that ruan s original crepant resolution conjecture should be modified by including appropriate quantum corrections and that there is no straightforward generalization of either ruan s original conjecture or the cohomological crepant resolution conjecture to the case of crepant partial resolutions our methods are based on mirror symmetry for toric orbifolds

structured procrustes problem abstract let s be a class of structured matrices given a pair of matrices x and b we consider the structured procrustes problem spp a arg min g s g x b f and provide a complete solution when s is either a jordan algebra or a lie algebra associated with an orthosymmetric scalar product we characterize and determine all solutions of the structured procrustes problem as well as those solutions which have the smallest norm we show that for the spectral norm there may be infinitely many smallest norm solutions of the structured procrustes problem whereas for the frobenius norm the smallest norm solution is unique

semi definite matrix constraints in optimization positive semi definite matrix constraints arise in a number of optimization problems in which some or all of the elements of a matrix are variables such as the educational testing and matrix modification problems the structure of such constraints is developed including expressions for the normal cone feasible directions and their application to optimality conditions a computational framework is given within which these concepts can be exploited and which permits the quantification of second order effects the matrix of lagrange multipliers in this formulation is shown to have an important relationship to the characterization of the normal cone modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem and comparative numerical experiments are described it is shown that a particular choice of the penalty parameter for an l 1 exact penalty function is appropriate for this type of problem the be

application of alternating convex projection methods for computation of positive toeplitz matrices uses alternating convex projection techniques to compute the closest positive definite toeplitz matrix that satisfies certain inequality constraints to a specified symmetric matrix some applications to signal processing and control problems are discussed

computing genus zero twisted gromov witten invariants twisted gromov witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold x which depend in addition on a vector bundle over x and an invertible multiplicative characteristic class special cases are closely related to local gromov witten invariants of the bundle and to genus zero one point invariants of complete intersections in x we develop tools for computing genus zero twisted gromov witten invariants of orbifolds and apply them to several examples we prove a quantum lefschetz theorem that expresses genus zero one point gromov witten invariants of a complete intersection in terms of those of the ambient orbifold x we determine the genus zero gromov witten potential of the type a surface singularity c2 zn we also compute some genus zero invariants of c3 z3 verifying predictions of aganagic bouchard and klemm 3 in a self contained appendix we determine the relationship between the quantum cohomology of the an surface singularity and that of its crepant resolution thereby proving the crepant resolution conjectures of ruan and of bryan and graber 12 in this case

inverse eigenvalue problem in structural design a method is presented for the design of a structure with specified low order natural frequencies the method can also be used for design problems in which certain bands devoid of any frequency constraints are specified this can further be used to generate initial feasible designs for optimum design problems with frequency constraints

left and right inverse eigenpairs problem of skew centrosymmetric matrices the left and right inverse eigenvalue problem of skew centrosymmetric matrices and its optimal approximation are considered new necessary and sufficient conditions for the solvability of the problem are derived the general expression of the solution is given the optimal approximate solution and some numerical experiments are also provided

inverse spectral problem for pseudo jacobi matrices with partial spectral data in this paper we study a special kind of matrices which are pseudo jacobi matrices four inverse eigenvalue problems are discussed and the necessary and sufficient conditions are presented under which these problems are solvable the corresponding algorithms and some examples are given numerical results demonstrate that these methods are of practical utility

hybrid methods for solving the educational testing problem abstract methods for solving the educational testing problem are considered one approach glunt 1995 is to formulate the problem as a linear convex programming problem in which the constraint is the intersection of three convex sets this method is globally convergent but the rate of convergence is slow however the method does have the capability of determining the correct rank of the solution matrix and this can be done in relatively few iterations if the correct rank of the solution matrix is known it is shown how to formulate the problem as a smooth nonlinear minimization problem for which a rapid convergence can be obtained by l1sqp method 6 this paper studies hybrid methods that attempt to combine the best features of both types of method an important feature concerns the interfacing of the component methods thus it has to be decided which method to use first and when to switch between methods difficulties such as these are addressed in the paper comparative numerical results are also reported

solutions with special structure to the linear matrix equation ax b in this article we solve the following three kinds of problems first some formulas for the minimal rank of the submatrices in a solution x of matrix equation ax b and the minimal and maximal rank of x itself are derived by using the matrix rank method from these formulas necessary and sufficient conditions are given for x to be nonsingular or the submatrices to be zero respectively second some formulas for the minimal rank of x x x x and the corresponding expressions of submatrices of x are investigated combined with matrix decomposition necessary and sufficient conditions are given for the existence of solutions to be hermitian local hermitian skew hermitian and local skew hermitian respectively third necessary and sufficient conditions are given for the existence of solutions to be local positive negative semidefinite and for some hermitian solution with zero submatrix the structure form of a positive negative semidefinite solution are obtained using results of inertia