minors of the moore penrose inverse abstract let q k n 1 k 1 1 k n denote the strictly increasing sequences of k elements from 1 n for q k n we denote by a the submatrix of a with rows indexed by columns by the submatrix obtained by deleting the rows and columns is denoted by a lsqb rsqb for nonsingular a r n n the jacobi identity relates the minors of the inverse a 1 to those of a det a 1 lsqb rsqb 1 k i 1 i k i 1 i det a lsqb rsqb det a for any q k n we generalize the jacobi identity to matrices a r m n r expressing the minors of the moore penrose inverse a in terms of the minors of the maximal nonsingular submatrices a ij of a in our notation det a lsqb rsqb 1 vol 2 a i j n det a ij a a ij for any q k m q k n 1 k r where vol 2 a denotes the sum of squares of determinants of r r submatrices of a we apply our results to questions concerning the nonnegativity of principal minors of the moore penrose inverse

a note on the representation and approximation of the outer inverse at s 2 of a matrix a in this note we present a unified representation and an approximation theorem for the outer inverse a t s 2 of a matrix a which is the matrix x satisfying xax x r x t n x s based on the new approximation theorem we can develop several expressions and computational procedures for the generalized inverse a t s 2 which improve the recent work on this topic

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

nilmanifolds of dimension 8 admitting anosov diffeomorphisms after more than thirty years the only known examples of anosov diffeomorphisms are topologically conjugated to hyperbolic automorphisms of infranilmanifolds and even the existence of an anosov automorphism is a really strong condition on an infranilmanifold any anosov automorphism determines an automorphism of the rational lie algebra determined by the lattice which is hyperbolic and unimodular and conversely these two conditions together are strong enough to make of such rational nilpotent lie algebras called anosov lie algebras very distinguished objects in this paper we classify anosov lie algebras of dimension less than or equal to 8 as a corollary we obtain that if an infranilmanifold of dimension n 8 admits an anosov diffeomorphism f and it is not a torus or a compact flat manifold i e covered by a torus then n 6 or 8 and the signature of f necessarily equals 3 3 or 4 4 respectively

moore penrose inverse of set inclusion matrices abstract given integers s k and v let w sk be the v s v k 0 1 matrix the rows and the columns of which are indexed by the s subsets and the k subsets of a v set respectively and where the entry in row s and column u is 1 if s u and 0 otherwise a formula for the moore penrose inverse of w sk over the reals is obtained a necessary and sufficient condition for w sk to admit a moore penrose inverse over the set of integers modulo a prime p is given together with a formula for the moore penrose inverse when it exists

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

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

lattice point problems and values of quadratic forms for d dimensional ellipsoids e with d 5 we show that the number of lattice points in re is approximated by the volume of re as r tends to infinity up to an error of order mathcal o r d 2 for general ellipsoids and up to an error of order o rd 2 for irrational ones the estimate refines earlier bounds of the same order for dimensions d 9 as an application a conjecture of davenport and lewis about the shrinking of gaps between large consecutive values of q m m d of positive definite irrational quadratic forms q of dimension d 5 is proved finally we provide explicit bounds for errors in terms of certain minkowski minima of convex bodies related to these quadratic forms

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

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

an oblique projection iterative method to compute drazin inverse and group inverse in this paper an oblique projection iterative method is presented to compute matrix equation axa a of a square matrix a with ind a 1 by this iterative method when taken the initial matrix x 0 a the group inverse a g can be obtained in absence of the roundoff errors if we use this iterative method to the matrix equation a kxa k a k a group inverse a k g of matrix a k is got then we use the formulae a d a k 1 a k g the drazin inverse a d can be obtained

on the generalized circle problem for a random lattice in large dimension abstract in this note we study the error term r n l x in the generalized circle problem for a ball of volume x and a random lattice l of large dimension n our main result is the following functional central limit theorem fix an arbitrary function f z r satisfying lim n f n and f n o e e e n for every e 0 then the random function t 1 2 f n r n l t f n on the interval 0 1 converges in distribution to one dimensional brownian motion as n the proof goes via convergence of moments and for the computations we develop a new version of rogers mean value formula from 18 for the individual kth moment of the variable 2 f n 1 2 r n l f n we prove convergence to the corresponding gaussian moment more generally for functions f satisfying f n o e c n for any fixed c 0 c k where c k is a constant depending on k whose optimal value we determine

some two step and three step nilpotent lie groups with small automorphism groups we construct examples of two step and three step nilpotent lie groups whose automorphism groups are small in the sense of either not having a dense orbit for the action on the lie group or being nilpotent the latter being stronger from the results we also get new examples of compact manifolds covered by two step simply connected nilpotent lie groups which do not admit anosov automorphisms

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

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

einstein solvmanifolds existence and non existence questions the aim of this paper is to study the problem of which solvable lie groups admit an einstein left invariant metric the space mathcal n of all nilpotent lie brackets on mathbb r n parametrizes a set of n 1 dimensional rank one solvmanifolds s mu mu in mathcal n containing the set of all those which are einstein in that dimension the moment map for the natural gl n action on mathcal n evaluated at mu in mathcal n encodes geometric information on s and suggests the use of strong results from geometric invariant theory for instance the functional on mathcal n whose critical points are precisely the einstein s s is the square norm of this moment map we use a gl n invariant stratification for the space mathcal n and show that there is a strong interplay between the strata and the einstein condition on the solvmanifolds s as an application we obtain criteria to decide whether a given nilpotent lie algebra can be the nilradical of a rank one einstein solvmanifold or not we find several examples of mathbb n graded even 2 step nilpotent lie algebras which are not a classification in the 7 dimensional 6 step case and an existence result for certain 2 step algebras associated to graphs are 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

the weighted moore penrose inverse of modified matrices the weighted moore penrose inverse of modified matrix a cb is derived under some range conditions where cb is a full rank factorization the generalized singular value decomposition of a matrix is used to derive systematically the weighted moore penrose inverse for a modified matrix where cb need not be a full rank factorization the results extend the earlier works by various authors

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

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

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

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

lattice point problems and distribution of values of quadratic forms for d dimensional irrational ellipsoids e with d 9 we show that the number of lattice points in re is approximated by the volume of re as r tends to infinity up to an error of order o rd 2 the estimate refines an earlier authors bound of order o rd 2 which holds for arbitrary ellipsoids and is optimal for rational ellipsoids as an application we prove a conjecture of davenport and lewis that the gaps between successive values say s 0 as s oo for d 9 for comparison note that sups n s s 0 for rational q x and d 5 as a corollary we derive oppenheim s conjecture for indefinite irrational quadratic forms i e the set q zd is dense in r for d 9 which was proved for d 3 by g margulis marl in 1986 using other methods finally we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums

rank factorization and bordering of regular matrices over commutative rings abstract let r be a commutative ring manjunatha prasad and bhaskara rao proved that every regular matrix over r can be completed to an invertible matrix of a particular size by bordering if and only if every regular matrix over r has a rank factorization and if and only if every finitely generated projective r module is free here we consider the case in which the bordering has no prescribed size and in which we take a rank factorization of a suitable extension of the given regular matrix for their prescribed size we discuss the existence of f r such that their borderings and their rank factorizations are true allowing f as a denominator

on the value distribution of the epstein zeta function in the critical strip we study the value distribution of the epstein zeta function e n l s for 0 infinity we prove that the random variable v n 2c e n cn has a limit distribution which we give explicitly here v n is the volume of the n dimensional unit ball more generally for any fixed epsilon 0 we determine the limit distribution of the random function c bar right arrow v n 2c e n cn c epsilon 1 4 epsilon 1 2 epsilon after compensating for the pole at c 1 2 we even obtain a limit result on the whole interval 1 4 epsilon 1 2 and as a special case we deduce the following strengthening of a result by sarnak and strombergsson concerning the height function h n l of the flat torus r n l the random variable n h n l log 4 pi gamma 1 log n has a limit distribution as n infinity which we give explicitly finally we discuss a question posed by sarnak and strombergsson as to whether there exists a lattice l subset of r n for which e n l s has no zeros in 0 infinity

characterizations approximation and perturbations of the core ep inverse abstract several characterizations and representations for the core ep inverse are presented and investigated to that goal several explicit bordering technique for characterizing and defining the core ep inverse are analyzed also the matrix splitting method and the iterative successive matrix squaring algorithm for computing the core ep inverse are investigated the perturbation bounds related with the core ep inverse are also estimated

on the distribution of lengths of short vectors in a random lattice we use an idea from sieve theory specifically an inclusion exclusion argument inspired by schmidt proc am math soc 9 390 403 1958 to estimate the distribution of the lengths of kth shortest vectors in a random lattice of covolume 1 in dimension n this is an improvement of the results of rogers proc lond math soc 6 3 305 320 1956 and sodergren math z 269 945 954 2011 in that it allows k to increase with n

anosov automorphisms on compact nilmanifolds associated with graphs we associate with each graph s e a 2 step simply connected nilpotent lie group n and a lattice in n we determine the group of lie automorphisms of n and apply the result to describe a necessary and sufficient condition in terms of the graph for the compact nilmanifold n f to admit an anosov automorphism using the criterion we obtain new examples of compact nilmanifolds admitting anosov automorphisms and conclude that for every n 17 there exist a n dimensional 2 step simply connected nilpotent lie group n which is indecomposable not a direct product of lower dimensional nilpotent lie groups and a lattice in n such that n f admits an anosov automorphism we give also a lower bound on the number of mutually nonisomorphic lie groups n of a given dimension satisfying the condition necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms

on surjectivity of the power maps of solvable lie groups abstract in this paper we study surjectivity of the map g g n on an arbitrary connected solvable lie group and describe certain necessary and sufficient conditions for surjectivity to hold the results are applied also to study the exponential maps of the lie groups

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

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

limit representations of generalized inverses and related methods we generalize the iterative method for implementation of the limit representation of the moore penrose inverse lim a email protected a a 1a introduced by zukovski and lipcer z vicisl mat i mat fiz 12 1972 843 15 1975 489 more precisely we define an iterative process for implementing the general limit formula lim a 0 ai r s 1r in a certain case the presented method gives an iterative process which implements the known limit representation of the drazin inverse introduced by meyer siam j appl math 26 1974 469 also the known limit representation of the generalized inverse a t s 2 is a special case of the general limit formula in the case of a full rank matrix a we define a limit representation and an iterative method for computation of the left or right inverses of a also we introduce two alternative limit representations for the set of 2 and 1 2 inverses and develop the corresponding iterative processes moreover we propose a finite algorithm for computation of the generalized inverses contained in the limit formula lim a 0 ai r s 1r using the method developed by ji appl math comput 61 1994 151

constructing infra nilmanifolds admitting an anosov diffeomorphism abstract in this paper we establish an algebraic characterization of those infra nilmanifolds modeled on a free c step nilpotent lie group and with an abelian holonomy group admitting an anosov diffeomorphism we also develop a new method for constructing examples of infra nilmanifolds having an anosov diffeomorphism

determinantal representations of the drazin inverse over the quaternion skew field with applications to some matrix equations abstract within the framework of the theory of the column and row determinants we obtain determinantal representations of the drazin inverse both for hermitian and arbitrary matrices over the quaternion skew field using the obtained determinantal representations of the drazin inverse we get explicit representation formulas analogs of cramer s rule for the drazin inverse solutions of a quaternion matrix equation axb d and consequently ax d and xb d in two cases if a b are hermitian or arbitrary

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

full rank representation of generalized inverse at s 2 and its application in this paper we introduce a full rank representation of the generalized inverse a t s 2 of a given complex matrix a which is based on an arbitrary full rank decomposition of g where g is a matrix such that r g t and n g s using this representation we introduce the minor of the generalized inverse a t s 2 as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained as an application we use an example to demonstrate that this representation is correct

moore penrose inverse in rings with involution abstract we study the moore penrose inverse mp inverse in the setting of rings with involution the results include the relation between regular mp invertible and well supported elements we present an algebraic proof of the reverse order rule for the mp inverse valid under certain conditions on mp invertible elements applications to c algebras are given

an alternative limit expression of drazin inverse and its application abstract we present an alternative limit expression of drazin inverse of a square matrix and a new proof for a finite algorithm for determining the drazin inverse is given using our new limit expression

successive matrix squaring algorithm for computing outer inverses abstract in this paper we derive a successive matrix squaring sms algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix a c r m n we generalize the results from the papers l chen e v krishnamurthy i macleod generalized matrix inversion and rank computation by successive matrix powering parallel computing 20 1994 297 311 y wei successive matrix squaring algorithm for computing drazin inverse appl math comput 108 2000 67 75 y wei h wu j wei successive matrix squaring algorithm for parallel computing the weighted generalized inverse a mn appl math comput 116 2000 289 296 and obtain an algorithm for computing various classes of outer generalized inverses of a instead of particular matrices used in these articles we use an appropriate matrix r c s n m s r numerical examples are presented

the image of the adjoint mapping abstract let r be a commutative ring with 1 and let a be an m n matrix over r with determinantal rank r 1 if 1 is in ideal of r generated by the r r minors of a then every inner inverse of a is in the image of a linear mapping r n r m r r n m that is specified in terms of the classical adjoints of the r r submatrices of a

examples of anosov diffeomorphisms we prove that if n is any graded rational lie algebra then the simply connected nilpotent lie group n n with lie algebra n r n r has a lattice such that the corresponding nilmanifold n n admits an anosov diffeomorphism this gives a procedure to construct easily several explicit examples of nilmanifolds admitting an anosov diffeomorphism and shows that a reasonable classification up to homeomorphism or even up to commensurability of such nilmanifolds would not be possible

automorphism invariant cartan subgroups and power maps of disconnected groups we extend a well known result of r steinberg on the existence of an invariant maximal torus under a semisimple automorphism of an algebraic group over an algebraically closed field we show that the same result holds when the underlying field is of characteristic zero but not necessarily algebraically closed we next study surjectivity of the power maps g mapsto g n of disconnected algebraic groups of characteristic zero in the case of disconnected real algebraic groups we apply our generalisation of steinberg s result to obtain results on the surjectivity of the power maps we also extend a result of a borel on weak exponentiality in real lie groups by relating it with the surjectivity of the square map

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

a finite algorithm for generalized inverses of polynomial and rational matrices there are proposed two types of finite algorithms for symbolic computation of outer inverses of a given polynomial or rational matrix based on the extension of the leverrier faddeev algorithm the implementation of the algorithm corresponding to rational matrices is developed in the symbolic package mathematica

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

solvable lie algebras and graphs abstract we define a solvable extension of the graph 2 step nilpotent lie algebras of 5 by adding elements corresponding to the 3 cliques of the graph we study some of their basic properties and we prove that two such lie algebras are isomorphic if and only if their graphs are isomorphic we also briefly discuss some metric properties providing examples of homogeneous spaces with nonpositive curvature operator and solvsolitons

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

neural network for computing pseudoinverses and outer inverses of complex valued matrices we propose two continuous time neural networks for computing generalized inverses of complex valued matrices with rank deficient cases the first of them is applicable in the pseudoinverse computation and the second one is applicable in construction of outer inverses the proposed continuous time neural networks have a low complexity of implementation and they are proved to be globally convergent without any condition compared with the existing algorithms for computing the pseudoinverse and outer inverses of matrices the global convergence of the proposed continuous time neural networks is analyzed in the complex domain effectiveness of the proposed continuous time neural networks is evaluated numerically via examples

computing 2 4 and 2 3 inverses by using the sherman morrison formula a finite recursive procedure for computing 2 4 generalized inverses and the analogous recursive procedure for computing 2 3 generalized inverses of a given complex matrix are presented the starting points of both introduced methods are general representations of these classes of generalized inverses these representations are formed using certain matrix products which include the moore penrose inverse or the usual inverse of a symmetric matrix product and the sherman morrison formula for the inverse of a symmetric rank one matrix modification the computational complexity of the methods is analyzed defined algorithms are tested on randomly generated matrices as well as on test matrices from the matrix computation toolbox

full rank representations of 2 4 2 3 inverses and successive matrix squaring algorithm abstract we present the full rank representations of 2 4 and 2 3 inverses with given rank as well as with prescribed range and null space as particular cases of the full rank representation of outer inverses as a consequence two applications of the successive matrix squaring sms algorithm from p s stanimirovic d s cvetkovic ilic successive matrix squaring algorithm for computing outer inverses appl math comput 203 2008 19 29 are defined using the full rank representations of 2 4 and 2 3 inverses the first application is used to approximate 2 4 inverses the second application after appropriate modifications of the sms iterative procedure computes 2 3 inverses of a given matrix presented numerical examples clarify the purpose of the introduced methods

symbolic computation of at s 2 inverses using qdr factorization abstract efficient evaluation of the full rank qdr decomposition is established a method and algorithm for efficient symbolic computation of a t s 2 inverses of a given rational matrix a is defined using the full rank qdr decomposition of an appropriate rational matrix w the algorithm is implemented using mathematica s ability to deal with symbolic expressions as well as with numbers examples including polynomial and rational matrices are presented some comparisons with well known methods for symbolic evaluation of generalized inverses are obtained

polynomial alternatives for the group of affine motions it is known that every polycyclic by finite group even if it admits no affine structure allows a polynomial structure of bounded degree a major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of mathbb r n in contrast to the group of affine motions is no longer a finite dimensional lie group in this paper we construct a family of finite dimensional lie groups even linear algebraic groups of polynomial diffeomorphisms which we call weighted groups of polynomial diffeomorphisms it turns out that every polycyclic by finite group admits a polynomial structure via these weighted groups in the nilpotent and other case s we can sharpen by specifying a nice set of weights the existence results obtained in earlier work we introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding mal cev completion this and other properties provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e g in the work of fried goldman and hirsch

on the poisson distribution of lengths of lattice vectors in a random lattice we prove that the volumes determined by the lengths of the non zero vectors x in a random lattice l of covolume 1 define a stochastic process that as the dimension n tends to infinity converg

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

computation of moore penrose generalized inverses of matrices with meromorphic function entries abstract in this paper given a field with an involutory automorphism we introduce the notion of moore penrose field by requiring that all matrices over the field have moore penrose inverse we prove that only characteristic zero fields can be moore penrose and that the field of rational functions over a moore penrose field is also moore penrose in addition for a matrix with rational functions entries with coefficients in a field k we find sufficient conditions for the elements in k to ensure that the specialization of the moore penrose inverse is the moore penrose inverse of the specialization of the matrix as a consequence we provide a symbolic algorithm that given a matrix whose entries are rational expression over c of finitely many meromorphic functions being invariant by the involutory automorphism computes its moore penrose inverve by replacing the functions by new variables and hence reducing the problem to the case of matrices with complex rational function entries

the structure of underlying lie algebras abstract every lie algebra over a field e gives rise to new lie algebras over any subfield f e by restricting the scalar multiplication this paper studies the structure of these underlying lie algebra in relation to the structure of the original lie algebra in particular the question how much of the original lie algebra can be recovered from its underlying lie algebra over subfields f by introducing the conjugate of a lie algebra we show that in some specific cases the lie algebra is completely determined by its underlying lie algebra furthermore we construct examples showing that these assumptions are necessary as an application we give for every positive n an example of a real 2 step nilpotent lie algebra which has exactly n different bi invariant complex structures this answers an open question by di scala lauret and vezzoni motivated by their work on quasi kahler chern flat manifolds in differential geometry the methods we develop work for general lie algebras and for general galois extensions f e in contrast to the original question which only considered nilpotent lie algebras of nilpotency class 2 and the field extension r c we demonstrate this increased generality by characterizing the complex lie algebras of dimension 4 which are defined over r and over q

surjectivity of power maps of real algebraic groups abstract in this paper we study the surjectivity of the power maps g g n for real points of algebraic groups defined over reals the results are also applied to study the exponentiality of such groups