on kineopsis and computation of structure and motion this correspondence deals with the computation of structure and motion of rigid objects in space from image positions and optical flow a test for rigid motion of objects in space is introduced which yields a new formulation of the problem assuming a central projection model for the viewing system it is shown that image positions and optical flow at four points can achieve this perception

a comparison of one dimensional traveling waves in inverse and normal fluidized beds abstract the state of uniform fluidization is usually unstable to small disturbances and this can lead to the formation of vertically traveling voidage waves in inverse fluidization when particle density is less than fluid density s f particles fluidize in the direction of gravity when the drag force exerted by the fluid overcomes buoyancy inverse fluidization thus provides a unique parameter space which augments the study of instability behavior in normal fluidization when f s using continuum equations of continuity and motion we compared the linear stability of normal and inverse bed modes to examine the effect of the froude number fr and fluid to solid density ratio f s making use of numerical bifurcation analysis and continuation periodic solutions in the form of one dimensional traveling waves 1d tws were computed based on wave growth rates and bifurcation structure we identified the fr as an important parameter for predicting instability strength however affects instability onset or the point at which the base state is rendered unstable in the case studies we examined traveling waves were shown to propagate in the direction of fluidization and asymmetrical high amplitude 1d tw profiles suggest fully developed bubble like structures are oriented in the direction of fluidization

generalized inverses over integral domains ii group inverses and drazin inverses this is a continuation of an earlier paper by the authors on generalized inverses over integral domains the main results consist of necessary and sufficient conditions for the existence of a group inverse a new formula for a group inverse when it exists and necessary and sufficient conditions for the existence of a drazin inverse we show that a square matrix a of rank r over an integral domain r has a group inverse if and only if the sum of all r r principal minors of a is an invertible element of r we also show that the group inverse of a when it exists is a polynomial in a with coefficients from r

bifurcation of plane voidage waves in fluidized beds abstract a two dimensional fluidized bed model is investigated from a bifurcation point of view it is shown that bifurcation to periodic travelling waves occurs when the basic state of uniform fluidization becomes unstable the nature of these waves may be one or two dimensional here the main focus is on the bifurcation of oblique travelling waves i e plane voidage waves propagating into various directions in space these waves can be described by a two dimensional dynamical system which can be analyzed extensively using phase plane methods several stationary and periodic solutions as well as homoclinic and heteroclinic orbits are shown to exist in the model studied

outer inverses and jacobi type identities abstract we consider matrices over a commutative ring and characterize the class of outer inverses for which jacobi type identities can be extended we obtain a necessary and sufficient condition for the existence of rao regular drazin inverse in terms of sum of principal minors of a k for some k also we obtain determinantal formula for the rao regular drazin inverse conjectures are formulated which give expressions for outer inverse and the conjectures are proved in some special cases

generalized inverses with proportional minors let a h be matrices of rank r and of order m n and n m respectively over an integral domain it is shown that a admits a g inverse whose r r minors are propotional to the corresponding minors of h if and only if the trace of the rth compound of ah is invertible we also obtain a determinantal formula for such a g inverse when it exists the results generalize earlier work on the existence of the moore penrose and group inverses

a comment on analysis of video image sequences using point and line correspondences abstract in this paper we would like to contest the results of y f wang n karandikar and j k aggarwal pattern recognition 24 1065 1085 1991 raising two fundamental claims a line does not contribute anything to recognition of motion parameters from two images four traceable points are not sufficient to recover motion parameters from two perspective projections to be constructive however we show that four traceable points are sufficient to recover motion parameters from two frames under orthogonal projection and that five points are sufficient to simplify the solution of the two frame problem under orthogonal projection to solving a linear equation system

outer inverses jacobi type identities and nullities of submatrices abstract according to the jacobi identity if a is an invertible matrix then any minor of a 1 equals up to a sign the determinant of a 1 times the complementary minor in the transpose of a the identity is extended to any outer inverse thereby generalizing several results in the literature for special generalized inverses a permanental analog of the jacobi identity is proved bounds are obtained for the difference between the nullity of a submatrix of a and that of the complementary submatrix in any generalized inverse or an outer inverse of a the result extends earlier work of fiedler markham and gustafson for the inverse and of robinson for the moore penrose inverse

full rank and determinantal representation of the drazin inverse abstract in this article we introduce a full rank representation of the drazin inverse a d of a given complex matrix a which is based on an arbitrary full rank decomposition of a l l k where k is the index of a using this general representation we introduce a determinantal representation of the drazin inverse more precisely we represent elements of the drazin inverse a d as a fraction of two expressions involving minors of the order rank a k k ind a taken from the matrices a and rank invariant powers a l l k also we examine conditions for the existence of the drazin inverse for matrices whose elements are taken from an integral domain finally a few correlations between the minors of the drazin inverse a d powers of the drazin inverse and the minors of the matrix a k k ind a are explicitly derived

drazin invertibility for matrices over an arbitrary ring abstract characterizations are given for existence of the drazin inverse of a matrix over an arbitrary ring moreover the drazin inverse of a product paq for which there exist a p and q such that p pa a aqq can be characterized and computed this generalizes recent results obtained for the group inverse of such products the results also apply to morphisms in additive categories as an application we characterize drazin invertibility of companion matrices over general rings

the moore penrose inverse of a morphism with factorization abstract given an m by n matrix a of rank r over a field with an involutory automorphism it is well known that a has a moore penrose inverse if and only if rank a a r rank aa by use of the full rank factorization theorem this result may be restated in the category of finite matrices as follows if a 1 r a 2 is an epic monic factorization of a m n through r then a has a moore penrose inverse if and only if a a 1 r a 2 and a 1 r a 2 a are respectively epic monic factorizations of a a n n and aa m m through r this characterization of the existence of moore penrose inverses is extended to arbitrary morphisms with epic monic factorizations

small froude number asymptotics in two dimensional two phase flows a nonlinear wave equation is derived describing the behavior of gas and liquid fluidized beds in the small froude number regime it represents a two dimensional perturbation of the korteweg char21 de vries equation and is shown to constitute a valid approximation of the original system while greatly simplifying the analytical and numerical investigation of two phase flow in fluidized beds it also leads to the conclusion that the underlying model does not significantly discriminate between gas and liquid fluidized beds near the stability limit an amplitude equation is derived governing the growth and stability of solitary plane waves the results are linked to those obtained by previous nonapproximative analyses it is expected that this analysis is applicable to other multiphase and traffic flow models due to the similarity in the governing equations and the completeness of the reduced wave equation

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

generalized invertibility of hankel and toeplitz matrices abstract a result for the computation of the moore penrose inverse of hankel matrices over the field of the complex numbers is given by heinig and rost in this note some methods of characterizing and computing generalized inverses of hankel and toeplitz matrices over fields and over rings with the extended rao condition are presented the results were obtained by combining classical theory on the special structure of those classes of matrices with recent theory on generalized invertibility

the generalized moore penrose inverse abstract we define the generalized moore penrose inverse and give necessary and sufficient conditions for its existence over an integral domain we also prove its uniqueness and give a formula for it which leads us towards a generalized cramer s rule to find the generalized moore penrose solution

motion estimation with more than two frames the development of a general motion estimation system based on using three or more frames is reported the relevant portion of the sequence is treated as a whole thus simplifying the process of estimating the motion parameters and making possible the computation of the motion parameters in terms of the natural center of motion a general description of this motion estimation method is presented with the derivation of motion equations for three cases three points in three frames two points in four frames and one point in five frames results are given for real and synthetic images for the case of one point in five and six frames on the average the program converged quickly to the right answer for noiseless data for noisy data the answers were reasonable and their accuracy was directly a function of the amount of noise in the disparity vector

determinantal representation of weighted generalized inverses abstract in this article we introduce new determinantal representations of the weighted generalized inverse by using a lemma of robinson d w robinson the classical adjoint linear algebra appl 411 2005 254 276 and using generalized cramer s rule we get the method for computing minors of the weighted moore penrose inverse and the w weighted drazin inverse

the moore penrose inverse of von neumann regular matrices over a ring abstract necessary and sufficient conditions are given in order that a von neumann regular matrix a over an arbitrary ring and also the matrices arising in certain factorizations of a be moore penrose invertible

generalized inverses of matrices over commutative rings abstract a rao regular matrix and the rao idempotent of a matrix over a commutative ring are defined we prove that a matrix a over a commutative ring is regular if and only if a is a sum of rao regular matrices with mutually orthogonal rao idempotents we find necessary and sufficient conditions for a matrix to have group inverse over a commutative ring also we give a method for computing minors of reflexive g inverse whenever it exists

regular hankel matrices over integral domains abstract in an earlier paper the author proved that a finite hankel matrix over a field has a rank factorization given by the extended frobenius theorem this paper states under what conditions such results can be extended to integral domains in such cases a characterization of regular hankel matrices is also given

generalized inverses of an invertible infinite matrix over a finite field in this paper we show that contrary to finite matrices with entries taken from finite field an invertible infinite matrix could have generalized inverses that are different from classical inverses

modules projectifs de type fini applications lineaires croisees et inverses generalises resume d une part nous developpons la theorie generale des inverses generalises de matrices en la mettant en rapport avec la theorie constructive des modules projectifs de type fini d autre part nous precisons certains aspects de cette theorie lies au calcul formel et a l analyse numerique matricielle nous demontrons en particulier qu on peut tester si un a module de presentation finie est projectif et calculer une matrice de projection correspondante en temps polynomial plus precisement pour une matrice a a m n on peut decider s il existe un inverse generalise b pour a c est a dire une matrice b verifiant a b a a et b a b b et en cas de reponse positive calculer un tel inverse generalise par un algorithme qui utilise o p 6 q 2 operations arithmetiques avec p inf m n q sup m n et un nombre polynomial de tests d appartenance d un element a un ideal engendre par un petit nombre d elements

a problem in computation of pseudoinverses in this paper we describe implementation of three methods for computing generalized inverses in the programming packages mathematica and delphi grevile s method the limit representation and the determinantal representation of generalized inverses are considered a major problem arising in the implementation of these methods is very rapid increment of arithmetic operations we describe a few different algorithms to avoid this common problem

stable two dimensional solitary pulses in linearly coupled dissipative kadomtsev petviashvili equations we present a two dimensional 2d generalization of the stabilized kuramoto sivashinsky system based on the kadomtsev petviashvili kp equation including dissipation of the generic newell whitehead segel nws type and gain the system directly applies to the description of gravity capillary waves on the surface of a liquid layer flowing down an inclined plane with a surfactant diffusing along the layer s surface actually the model is quite general offering a simple way to stabilize nonlinear media combining the weakly 2d dispersion of the kp type with gain and nws dissipation other applications are internal waves in multilayer fluids flowing down an inclined plane double front flames in gaseous mixtures etc parallel to this weakly 2d model we also introduce and study a semiphenomenological one whose dissipative terms are isotropic rather than of the nws type in order to check if qualitative results are sensitive to the exact form of the lossy terms the models include an additional linear equation of the advection diffusion type linearly coupled to the main kp nws equation the extra equation provides for stability of the zero background in the system thus opening a way for the existence of stable localized pulses we focus on the most interesting case when the dispersive part of the system is of the kp i type which corresponds e g to capillary waves and makes the existence of completely localized 2d pulses possible treating the losses and gain as small perturbations and making use of the balance equation for the field momentum we find that the equilibrium between the gain and losses may select two steady state solitons from their continuous family existing in the absence of the dissipative terms the latter family is found in an exact analytical form and is numerically demonstrated to be stable the selected soliton with the larger amplitude is expected to be stable direct simulations completely corroborate the analytical predictions for both the physical and phenomenological models

i inverness for polynomial matrices of non constant rank abstract let p1 pt be polynomials in c n with a variety v of common zeros contained in a suitable open set u explicit formulas are provided to construct rational functions 1 s such that i 1spi i 1 and such that the singularities of the i are contained in u this result is applied to compute rational functions valued 1 inverses of matrices with polynomial coefficients which do not have constant rank while retaining control over the location of the singularities of the rational functions themselves

on explicit solutions to the bezout equation explicit constructions are given for the solution of the bezout equation for matrices whose coefficients are multivariate polynomials or entire functions of paley wiener type this procedure also provides the 1 inverse when the bezout equation has no solutions

on bordering of regular matrices necessary and sufficient conditions are given for a commutative ring r to be a ring over which every regular matrix can be completed to an invertible matrix of a particular size by bordering such rings are precisely the projective free rings also over such rings every regular matrix has a rank factorization using the bordering technique we give an interesting method of computing minors of a reflexive g inverseg of a regular matrix a when i ag and i ga have rank factorizations

nonlinear waves in two fluid flows abstract typically the interaction of two miscible fluids leads to the formation of segregation patterns in the form of plane concentration waves and two or three dimensional voids or clusters the development of small amplitude plane travelling waves and the subsequent onset of higher dimensional instabilities can be elucidated analytically in addition numerical methods allow the computation of higher amplitude solutions as both direct approaches become costly although at different amplitude levels one is tempted to look for a simpler model problem therefore a nonlinear wave equation has been derived for two dimensional dispersed two phase flow in fluidized beds near the stability boundary using the froude number as small parameter this equation is a two dimensional perturbation of the korteweg de vries equation containing all low order terms it allows to identify the transition point between voids and clusters and to investigate these two regions for solitary and periodic waves a ginzburg landau equation is derived to describe transverse perturbations of one dimensional wavetrains bifurcating at the stability boundary

some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive image frames this paper describes the results of some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive tv images and discusses several factors which affect the accuracy of the results these factors include the sizes of the motion parameters themselves the accuracy of the raw data and the number of point correspondences in addition we address two related topics determining corner positions to subpixel accuracy and matching point patterns with different scales

the moore idempotents of a matrix abstract let r be a commutative ring with 1 and with an involution and let mr be the category of finite matrices over r with the involution aij a ij a ji if a is in mr then there exists a unique list e0 es of pairwise orthogonal symmetric idempotents of r that are characterized by several properties involving the rank and squared volume of the eia in particular a has a moore penrose inverse in mr iff esa 0

symmetric morphisms and the existence of moore penrose inverses abstract let x y be a morphism of an additive category with an involution then has a moore penrose inverse with respect to iff is invertible for some annihilator n x of iff there exists an object z and morphisms x z z z z y such that 2 and 1 z and 1 z are invertible in this case 1 1 z 1 1 z 1

the classical adjoint abstract this paper summarizes the historical background of the notion of the classical adjoint as outlined by muir and provides applications of the adjoint to various studies of generalized invertibility of matrices over commutative rings specifically in this setting the classical adjoint is used to provide a novel proof of von neumann s 1936 observation that every matrix over a regular ring is regular and to provide a necessary and sufficient condition for the existence of the moore penrose inverse of a given matrix in particular a representation of the moore penrose inverse is given that leads to an immediate proof of moore s 1920 formula specifying the entries of his reciprocal in terms of determinants

nonlinear waves and solitons in physical systems abstract advances in nonlinear science have been plentiful in recent years in particular interest in nonlinear wave propagation continues to grow stimulated by new applications such as fiber optic communication systems as well as the many classical unresolved issues of fluid dynamics what is arguably the turning point for the modern perspective of nonlinear systems took place at los alamos over 40 years ago with the pioneering numerical simulations of fermi pasta and ulam a decade later this research initiated the next major advance of zabusky and kruskal that motivated the revolution in completely integrable systems with this in mind the conference on nonlinear waves in solitions in physical systems nwsps was organized by the center for nonlinear studies cnls at los alamos national laboratory in may of 1997 to assess the current state of the art in this very active field papers from the conference attendees as well as from researchers unable to attend the conference were collected in this special volume of physica d in this paper the contributions to the conference and to this special issue are reviewed with an emphasis on the many unifying principles that all these works share

a volume associated with m x n matrices abstract let a mxnr with nonzero singular values 1 2 r the volume of a vol a is defined as zero if r 0 and otherwise vola ri 1 i or equivalently vola det2aij summing over all r x r nonsingularsubmatrices aij the matrix volume vol a generalizes the absolute value of the determinant from nonsingular to arbitrary matrices any r dimensional unit cube in r at is mapped under a into a parallelepiped of volume vol a this paper gives properties and applications of vol a in particular the moore penrose inverse of a is a convex combination of ordinary inverses of its maximal nonsingular submatrices a i j ij or is a convex combination of moore penrose inverses of its maximal full rank submatrices a i i j j where the convex weights are proport ional to the squares of volumes of the corresponding submatrices since a a the matrix a is a convex combination of inverses of submatrices of a interestingly the weights of these convex combinations are the above weights in transposed position

strong selection or rejection of spatially periodic patterns in degenerate bifurcations abstract when studying bifurcating space periodic solutions one often has a situation common in many problems of fluid dynamics in which in the r k plane r a control parameter such as reynolds number k wave number the passage from linear stability to instability is characterized by a parabola like curve having a minimum at r c k c we analyse classes of problems in which the coefficients of nonlinear terms in the amplitude equation go through zero near k c various examples of such degenerate problems can be found in the literature but have not been studied yet we give an extensive classification of bifurcation pictures which display rather unusual behaviour we further study the stability of these periodic solutions subject to quite general perturbation we find as a general result that all bifurcating solutions are unstable except for a small neighbourhood of a curve in the r k plane there is hence a strong selection mechanism of periodic patterns which fixes for each value of r with a very small uncertainty the wave number of the stable bifurcating periodic solution for one class of problems the curve continues to exist for all values of r for which our theory is consistent for another class the curve stops at a value r r c for r r all bifurcating periodic solutions are unstable in this case instability provides a mechanism of rejection of all periodic patterns in the last section we analyse the particular case of the blasius boundary layer flow

estimating three dimensional motion parameters of a rigid planar patch we present a new direct method of estimating the three dimensional motion parameters of a rigid planar patch from two time sequential perspective views image frames first a set of eight pure parameters are defined these parameters can be determined uniquely from the two given image frames by solving a set of linear equations then the actual motion parameters are determined from these pure parameters by a method which requires the solution of a sixth order polynomial of one variable only and there exists a certain efficient algorithm for solving a sixth order polynomial aside from a scale factor for the translation parameters the number of real solutions never exceeds two in the special case of three dimensional translation the motion parameters can be expressed directly as some simple functions of the eight pure parameters thus only a few arithmetic operations are needed

new additive results for the generalized drazin inverse in this paper we investigate additive properties of generalized drazin inverse of two drazin invertible linear operators in banach spaces under the commutative condition of pq qp we give explicit representations of the generalized drazin inverse p q d in term of p pd q and qd we consider some applications of our results to the perturbation of the drazin inverse and analyze a number of special cases

analysis of video image sequences using point and line correspondences abstract in this research we investigate the problem of analyzing time varying imagery using a feature based approach we assume a scenario where the imaged objects remain stationary while the camera moves the goal is to compute the structure of the imaged objects and the motion of the camera from a sequence of video images the proposed method exploits the principle of the invariance of rigid configurations during motion using the rigidity constraints we specify equations based on distance and angular invariance to compute the structural parameters of the imaged 3d objects independently of the camera s motion once the structural parameters are recovered the motion parameters can be computed the strength of our approach is in the decomposition of the computations of structure and motion and in the use of point and line correspondences simultaneously the main advantage of using both points and lines is that our approach is not limited to objects whose images have only a particular type of feature in abundance in this research we consider the special case of four points and one line which are the minimum feature sets required for computing the structure and motion parameters we then consider additional features in an overdetermined system of equations to improve the reliability and accuracy of the computation we present computer simulation results as well as results from real image sequences to demonstrate the algorithm s validity

the determinal rank idempotents of a matrix abstract let m r be the category of finite matrices over a commutative ring r with 1 for a m r of determinantal rank r let i r a be the ideal of r generated by the r r minors of a there exists a unique list e 1 e t of pairwise orthogonal idempotents of r that sum to 1 such that if r i rank e i a then rank a r 1 r 2 r t e i is the identity element of i r i e i a for 1 i t and either e t a 0 or i r t e t a does not possess an identity element characterizations are given for various generalized inverses of a e 1 a e t a in particular a is regular iff e t a 0

group and moore penrose inverses of regular morphisms with kernel and cokernel abstract let y be a morphism with kernel k x and cokernel y l in an additive category c 1 if x y then has a group inverse iff is invertible and is regular 2 if c has an involution then has a moore penrose inverse with respect to iff and are invertible and is regular

generalized inverses over integral domains in an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r r minors of the matrix is one in the same paper a procedure for constructing a generalized inverse from such a linear combination was also given in the present paper we show that any reflexive generalized inverse can be obtained by that procedure we also show that if the integral domain is a principal ideal domain every generalized inverse can be obtained by that procedure it is also shown that a matrix a of rank r over an integral domain has moore penrose inverse if and only if the sum of squares of all r r minors of a is an invertible element of the integral domain

on generalized inverses of matrices over integral domains it is proved that a matrix a over an integral domain admits a 1 inverse if and only if a linear combination of all the r r minors of a is equal to one where r is the rank of a some results on the existence of moore penrose inverses are also obtained

on a constraint equation for the estimation of displacement rates in image sequences the commonly used constraint equation delta g sup t u g sub t 0 for the estimation of optical flow can only be justified by assumptions that are in general far too restrictive for image sequences of real world scenes b g schunck 1985 86 recently argued that a constraint equation for the estimation of what he called image flow has to include a term containing the divergence of this image flow without presenting however a stringent derivation based on perspective projection from 3 d scene space the present author derives a constraint equation based on a combination of perspective projection and notions from differential geometry in addition he demonstrates the quantitive effects of taking into account radiometric considerations based on the use of lambertian reflection properties and isotropic illumination in scene space

diagonalisation of idempotent matrices let r be a ring associative with unit element and with an involution on it an m x n matrix a is said to have a moore penrose mp inverse with respect to the involution iff there exists an n x m matrix x such that axa a x 4x x ax ax xa xa the solution if it exists is unique and denoted by a several authors considered the problem of characterising matrices over certain domains for which an mp inverse exists cf 1 3 61 there results were generalised by puystjens and robinson cf 4 the latter noted that if an m x iz matrix a over a ring is of the form

using constancy of distance to estimate position and displacement in space the problem of computing structure and motion from the observation of points in two distinct images of a scene is considered the proposed method explicitly utilizes the principle of conservation of distance during rigid body motion the formulation is such that it separates the problem of estimating object position from that of determining motion parameters the equations of invariance of distance for a rigid body are solved for the points position in space when these coordinates in space are known motion parameters are computed in a simple and straightforward manner examples are given to illustrate the efficiency of the algorithm

categories of matrices with only obvious moore penrose inverses abstract let r be an associative ring with 1 and with an involution a and let mr be the category of finite matrices over r with the involution aij aij ji then the following two statements are equivalent i if a in mr has a moore penrose inverse with respect to then a is permutationally equivalent to a matrix of the form b 0 0 0 with b invertible ii if 1 a in r then at most one of the a s is not zero

module theoretic approach to controllability of convolutional systems both the behavioral approach proposed by willems and the module theoretic approach proposed by fliess provide a representation free theory of dynamical systems according to these approaches controllability is an intrinsic system property independent of the system representation in the first part of this paper a comparison between the behavioral approach and the module theoretic approach to convolutional systems is proposed then controllability of this class of systems is analyzed according to these two approaches this concept is related to the existence of image representations finally the same analysis is proposed for systems described by delay differential equations

about the von neumann regularity of triangular block matrices abstract necessary and sufficient conditions are given for the von neumann regularity of triangular block matrices with von neumann regular diagonal blocks over arbitrary rings this leads to the characterization of the von neumann regularity of a class of triangular toeplitz matrices over arbitrary rings some special results and a new algorithm are derived for triangular toeplitz matrices over commutative rings finally the drazin invertibility of some companion matrices over arbitrary rings is considered as an application

outer inverses of matrices abstract let r be a commutative ring with 1 and let g r n m if a r m n is of rank r 1 with tr c r ag 1 then aga a iff gcr a a that is a is an outer inverse of g iff a is the adjoint image of cr a under g

the moore penrose inverse over a commutative ring abstract let r be a commutative ring with 1 and with an involution a and let mr be the category of finite matrices over r with the involution a ij a ij a ji a matrix a m n in mr of determinantal rank r such that u a q r m q r m det a det a has a moore penrose inverse u a in r is said to be moore invertible with moore idempotent u a u a if u a u a a a for every matrix a of mr a has a moore penrose inverse with respect to if and only if a is the sum of moore invertible matrices whose moore idempotents are pairwise orthogonal