the congruence class of the solutions of some matrix equations the congruence class of a least square solution for the following matrix equations ax b a xa d axb dand axxb ef is presented also we derive necessary and sufficient conditions for the existence of a least square solution and present a general form of such solutions using the singular value decomposition svd and generalized singular value decomposition gsvd

on g inverses of a bordered matrix revisited abstract in this article we study the structure and properties of g inverse m of the bordered matrix m a b c 0 which is partitioned conforming with m m d 1 d 2 d 3 d 4 by applying the multiple quotient singular value decomposition qq svd from the structure of m we discuss the relations among d1 d2 d3 and d4 and derive equivalent conditions for m to be a bordered matrix by studying d1 in m we obtain some equivalent conditions for d1 to be a g inverse of a

reverse order law for reflexive generalized inverses of products of matrices abstract the relationships between generalized inverses of the product of two matrices a b and the product of generalized inverses of a b have been studied in the literature by applying the product singular value decomposition p svd in this paper we derive equivalent conditions for b 1 2 a 1 2 ab 1 2 ab 1 2 b 1 2 a 1 2 b 1 2 a 1 2 ab 1 2 and equivalent conditions for ab 1 b 1 a 1

on the early history of the singular value decomposition this paper surveys the contributions of five mathematicians eugenio beltrami 1835 1899 camille jordan 1838 1921 james joseph sylvester 1814 1897 erhard schmidt 1876 1959 and hermann weyl 1885 1955 who were responsible for establishing the existence of the singular value decomposition and developing its theory

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

augmented hooke s law in frequency domain a three dimensional material damping formulation abstract an isothermal fully three dimensional material damping modelling technique to some extent alternative to classic viscoelasticity is proposed the method is formulated in the frequency domain as an augmented hooke s law ahl with a constitutive matrix in which material damping is introduced by adding frequency dependent complex valued terms to the classical material modulus matrix of hooke s generalized law the derivations are based on linear irreversible thermodynamics and the concept of hidden coordinates as introduced by biot 1955 phys rev 97 1463 1469 the one dimensional concept of multiple augmenting thermodynamic fields by lesieutre 1992 int j solids structures 29 1567 1579 is generalized to a suitable three dimensional continuum form through the introduction of a special type of hidden coordinate vectors with linear first order time domain relaxation equations consistent with the hidden coordinates a free energy density function assuming isothermal conditions is introduced as the time domain basis of the augmented hooke s law through a time domain model of the coupled evolution of the mechanical displacements and the thermodynamical variables issues of causality are avoided completely in the final frequency domain formulation the general time domain model used is shown to be equivalent to a three dimensional multiple anelastic displacement field model an isotropic augmented hooke s law with both dilatational and shearing damping has been implemented and tested using a 20 node volume element in the finite element code aska acoustics a close agreement between finite element calculations and the corresponding analytically exact results for the studied rod and beam cases is obtained

transient response analysis of systems with different damping models a response of systems with different damping models to transient excitations is studied using the finite element method for this purpose an efficient method using fast fourier transform has been developed this method is based on the trigonometrical representation of the input signals and matrix of the transfer functions as an example systems with viscous and viscoelastic structural and external damping are examined two forms of viscous damping are available in the present implementation raleigh damping and element damping the viscoelastic material behaviour is represented by the complex modulus model the present implementation gives the possibility to preserve exactly the frequency dependence of the storage and loss moduli of viscoelastic materials logarithmic decrements are determined using the steady state vibrations of systems to characterize their damping properties test problems and numerical examples are given to demonstrate the validity and application of the approach suggested in this paper

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

simultaneous decomposition of quaternion matrices involving ź hermicity with applications let r and h m n stand respectively for the real number field and the set of all m n matrices over the real quaternion algebra h a 0 a 1 i a 2 j a 3 k i 2 j 2 k 2 ijk 1 a 0 a 1 a 2 a 3 r for i j k a real quaternion matrix a h n n is said to be hermitian if a a where a a and a stands for the conjugate transpose of a arising in widely linear modeling we present a simultaneous decomposition for a set of nine real quaternion matrices involving hermicity with compatible sizes a i h p i t i b i h p i t i 1 and c i h p i p i where ci are hermitian matrices i 1 2 3 as applications of the simultaneous decomposition we give necessary and sufficient conditions for the existence of an hermitian solution to the system of coupled real quaternion matrix equations a i x i a i b i x i 1 b i c i i 1 2 3 and provide an expression of the general hermitian solutions to this system moreover we derive the rank bounds of the general hermitian solutions to the above mentioned system using ranks of the given matrices ai bi and ci as well as the block matrices formed by them finally some numerical examples are given to illustrate the results of this paper

a simultaneous decomposition for seven matrices with applications abstract let h m n be the set of all m n matrices over the quaternion algebra h in this paper we construct a simultaneous decomposition for seven matrices with compatible sizes a h m n b h m p 1 c h m p 2 d h m p 3 e h q 1 n f h q 2 n and g h q 3 n as applications of the simultaneous matrix decomposition we give some solvability conditions general solutions as well as the range of ranks of the general solutions to the following two generalized sylvester matrix equations b x e c y f d z g a and b x w e c y f d z g a where a b c d e f and g are given quaternion matrices moreover we provide some numerical examples to illustrate the results of this paper

differential versus integral formulation of fractional hyperviscoelastic constitutive laws for brain tissue modelling for years interest has been constantly growing in biological tissue modelling particularly the mechanical study of the brain has become a major topic in the field of biomechanics a global model of this organ including a realistic mesh and suitable constitutive laws for the different tissues would find applications in various domains such as neurosurgery haptic device design or car manufacturing to evaluate the possible trauma due to an impact several constitutive models have already been designed regarding the strong strain rate dependence of the stress strain curves available in the literature we decided to describe brain tissue as a viscoelastic medium through the use of the fractional derivation operator thanks to this approach we can derive a convolution based model with the mittag leffler function as the regularized kernel

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

time domain finite element analysis of viscoelastic structures with fractional derivatives constitutive relations numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional calculus operators are presented to avoid difficulties concerning fractional order initial conditions a form of the fractlonal calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of mittag lefler type is used the constitutive equation is generalized to three dimensional states for isotropic materials a simplification of the fractional derivative of the memory kernel is used in connection with grunwald s definition of fractional differentiation and a backward euler rule for the time evolution of the convolution term a desirable feature of this process is that no actual evaluation of the memory kernel is needed this together with the newmark method for time integration enables the direct calculation of the time evolution of the nodal degrees of freedom to illustrate the ability of the numerical procedure a few numerical examples are presented in one example the numerically obtained solution is compared with a time series expansion of the analytical solution

a note on the reverse order law for reflexive generalized inverse of multiple matrix products the reverse order law for reflexive generalized inverses of multiple matrix products was introduced and discussed in m wei reverse order laws for generalized inverse of multiple matrix products linear algebra appl 293 1999 273 288 there the author derived some necessary and sufficient conditions for the reverse order lawa n 1 2 a n 1 1 2 a 1 1 2 a 1a 2 a n 1 2 by applying the product singular value decomposition p svd in this note we revisited this reverse order law by using the extremal rank relations of generalized schur complements and a new simpler equivalent condition is obtained in terms of only the ranks of the known matrices

multi parameter regularization techniques for ill conditioned linear systems when a system of linear equations is ill conditioned regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties the ill conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen in this paper we consider the case of several regularizations terms added simultaneously thus overcoming the problem of the best choice of the regularization matrix the error of this procedure is analyzed and numerical results prove its efficiency

an adaptation of the gear scheme for fractional derivatives abstract the gear scheme is a three level step algorithm backward in time and second order accurate for the approximation of classical time derivatives in this contribution the formal power of this scheme is proposed to approximate fractional derivative operators in the context of finite difference methods some numerical examples are presented and analysed in order to show the effectiveness of the present gear scheme at the power g scheme when compared to the classical grunwald letnikov approximation in particular for a fractional damped oscillator problem the combined g newmark scheme is shown to be second order accurate

reverse order laws for least squares g inverses and minimum norm g inverses of products of two matrices in this paper we derive equivalent conditions for reverse order laws of least squares g inverses and minimum norm g inverses of product of two matrices a and b by applying the product singular value decomposition p svd

two sided coupled generalized sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices abstract in this paper we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices a i c p i t i b i c s i q i c i c p i t i 1 d i c s i 1 q i and e i c p i q i i 1 2 3 we show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two sided coupled generalized sylvester matrix equations with four unknowns a i x i b i c i x i 1 d i e i i 1 2 3 apart from proving an expression for the general solutions to this system we derive the range of ranks of these solutions using the ranks of the given matrices a i b i c i d i and e i we provide some numerical examples to illustrate our results moreover we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two sided coupled generalized sylvester matrix equations with k 1 unknowns a i x i b i c i x i 1 d i e i i 1 k k 4 the main results are also valid over the real number field and the real quaternion algebra

equivalent conditions for generalized inverses of products abstract the relationships between generalized inverses of the product of two matrices a b and the product of generalized inverses of a b have been studied in the literature and equivalent conditions for b 1 a 1 ab 1 have been reported by applying the product singular value decomposition we derive more equivalent conditions for b 1 a 1 ab 1 and derive equivalent conditions for b 1 a 1 ab 1

on least squares g inverses and minimum norm g inverses of a bordered matrix abstract by applying the restricted singular value decomposition r svd qq svd in this article we derive structures of least squares g inverses and minimum norm g inverses of a bordered matrix m a b c 0 we also obtain the specific formulation of the pesudo inverse m

reverse order laws for generalized inverses of multiple matrix products abstract in this article we study reverse order laws for generalized inverses and reflexive generalized inverses of the products of multiple matrices a 1 a n and the products of generalized inverses and reflexive generalized inverses of a n a 1 by applying the multiple product singular value decomposition we obtain necessary and sufficient conditions for one side inclusion relation of reverse order law for generalized inverses and necessary and sufficient conditions of reverse order law for reflexive generalized inverses

time domain modeling of damping using anelastic displacement fields and fractional calculus abstract a fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed the evolution equation for the anelastic part is then a differential equation of fractional order in time by using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics to illustrate the modeling capability the model parameters are fit to available frequency domain data for a high damping polymer by studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning the use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined including the treatment of boundary conditions the anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of mittag leffler function type finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented

extremal ranks of matrix expression of a bxc with respect to hermitian matrix abstract the extremal ranks of matrix expression of a bxc with respect to x h x have been discussed by applying the quotient singular value decomposition q svd and some rank equalities of matrices in this paper

statistical origins of fractional derivatives in viscoelasticity many linear viscoelastic materials show constitutive behavior involving fractional order derivatives linear time invariant systems without memory have exponential decay in time but contradictorily not the power law decay associated with fractional derivatives the physics literature has noted that apparently non exponential decays can be observed when several simultaneously decaying processes have closely spaced exponential decay rates many engineer researchers interested in viscoelastic damping however seem unaware of these observations in this letter i give an unoriginal explanation but with a fresh engineering flavor for the appearance of these fractional order derivatives by this explanation fractional order damping can be expected from many materials with sufficiently disordered dissipation mechanisms

on the structure and geometry of the product singular value decomposition the product singular value decomposition is a factorization of two matrices which can be considered as a generalization of the ordinary singular value decomposition at the same level of generality as the quotient generalized singular value decomposition a constructive proof of the product singular value decomposition is provided which exploits the close relation with a symmetric eigenvalue problem several interesting properties are established the structure and the non uniqueness properties of the so called contragredient transformation which appears as one of the factors in the product singular value decomposition are investigated in detail finally a geometrical interpretation of the structure is provided in terms of principal angles between subspaces

computing the generalized singular values vectors of large sparse or structured matrix pairs we present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair a b the algorithm is based on the cs decomposition and the lanczos bidiagonalization process at each iteration step of the lanczos process the solution to a linear least squares problem with a rm t b rm t rm t as the coefficient matrix is approximately computed and this consists the only interface of the algorithm with the matrix pair a b numerical results are also given to demonstrate the feasibility and efficiency of the algorithm

forward order law for g inverses of the product of two matrices abstract in this paper we derive equivalent conditions for a b ab by the method of maximal rank and thus we obtain equivalent conditions for the forward order laws of the product of two matrices

nonlinear vibration of viscoelastic beams described using fractional order derivatives abstract the problem of non linear steady state vibration of beams harmonically excited by harmonic forces is investigated in the paper the viscoelastic material of the beams is described using the zener rheological model with fractional derivatives the constitutive equation which contains derivatives of both stress and strain significantly complicates the solution to the problem the von karman theory is applied to take into account geometric nonlinearities amplitude equations are obtained using the finite element method together with the harmonic balance method and solved using the continuation method the tangent matrix of the amplitude equations is determined in an explicit form the stability of the steady state solution is also examined a parametric study is carried out to determine the influence of viscoelastic properties of the material on the beam s responses

formulation and integration of the standard linear viscoelastic solid with fractional order rate laws abstract a physically sound three dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented it is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of mittag leffler function type a time integration scheme based on the generalized midpoint rule together with the grunwald algorithm for numerical fractional differentiation for integration of the constitutive response is developed the predictive capability of the viscoelastic model for describing creep relaxation and damped dynamic responses is investigated both analytically and numerically the algorithm and the present general linear viscoelastic model are implemented into the general purpose finite element code abaqus the algorithm is then used together with an explicit difference scheme for integration of structural responses in numerical examples the quasi static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated

analytical solution of a convection dispersion model with time dependent transport coefficients mathematical studies of solute transport in porous media have often utilized equivalent models of the transport process to remove undesired variability in the transport coefficients at the space and time scales of direct interest both deterministic and stochastic approaches in this genre produce an effective convection dispersion equation with time dependent coefficients this type of equation in one spatial dimension is investigated mathematically in the present paper a closed form solution of the solute transport equation is derived for a semi infinite spatial domain with arbitrary initial and boundary flux conditions it is shown that the solution reduces to well known results for special forms of the time dependent coefficients in general however a volterra integral equation of the second kind must be solved to evaluate the analytical solution of the transport equation we present a stable and convergent numerical scheme utilizing a trapezoidal quadrature rule for the solution of the volterra equation the method of solution developed should be applicable to a broad variety of solute transport problems including particularly those in heterogeneous porous media