products of projections and positive operators this article is devoted to the study of the set t of all products pa with p an orthogonal projection and a a positive semidefinite operator we describe this set and study optimal factorizations we also relate this factorization with the notion of compatibility and explore the polar decomposition of the operators in t

the center of the generic division ring and twisted multiplicative group actions abstract let f be a field and let p be a prime the problem we study is whether the center cp of the division ring of p p generic matrices is stably rational over f given a finite group g and a z g lattice we let f m be the quotient field of the group algebra of the abelian group m procesi and formanek linear multilinear algebra 7 1979 203 212 have shown that for all n there is a z s n lattice gn such that cn is stably isomorphic to the fixed field under the action of sn of f gn let h be a p sylow subgroup of sp let a be the root lattice and let l f z s p h we show that there exists an action of sp on l z s p z h a twisted by an element ext 1 s p z s p z h a l such that l z s p z h a s p is stably isomorphic to cp the extension corresponds to an element of the relative brauer group of l over lh since z s p z h a and z s p h are quasi permutations l z s p z h a s p is stably rational over f however it is not known whether l z s p z h a s p is stably rational over f thus the result represents a reduction on the problem since z s p z h a is quasi permutation however the twist introduces a new level of complexity

construction of hadamard mathbb z _ 2 mathbb z _ 4 q _ 8 codes for each allowable value of the rank and dimension of the kernel this paper deals with hadamard mathbb z 2 mathbb z 4 q 8 codes which are binary codes after a gray map from a subgroup of direct products of mathbb z 2 mathbb z 4 and q 8 where q 8 is the quaternionic group in a previous work these codes were classified in five shapes in this paper we analyze the allowable range of values for the rank and dimension of the kernel which depends on the particular shape of the code we show that all these codes can be represented in a standard form from a set of generators which can help in understanding the characteristics of each shape the main results we present are the characterization of hadamard mathbb z 2 mathbb z 4 q 8 codes as a quotient of a semidirect product of mathbb z 2 mathbb z 4 linear codes and the construction of hadamard mathbb z 2 mathbb z 4 q 8 codes with each allowable pair of values for the rank and dimension of the kernel

on hadamard property of a certain class of finite groups abstract utilizing results of nekrasov and berkovich we investigate hadamard property of a certain class of finite groups

on the convergence of linear switched systems this paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems we provide a collection of results that use weak dwell time dwell time strong dwell time permanent and persistent activation hypothesis the obtained results are shown to be tight by counterexample finally we apply our result to the three cell converter

the rate of convergence for the method of alternating projections ii abstract the purpose of the paper is threefold 1 to develop a useful error bound for the method of alternating projections which is relatively easy to compute and remember 2 to exhibit a counterexample to a conjecture of kayalar and weinert 3 to show that in the case of at least three subspaces any error bound which only depends on the angles between the various subspaces involved can never be sharp

the numerical range and the spectrum of a product of two orthogonal projections the aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum several improvements removing the closure of the numerical range of the operator using a parametrization after its eigenvalues are possible under additional assumptions an estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the friedrichs angle applications to the rate of convergence in the method of alternating projections and to the uncertainty principle in harmonic analysis are also discussed

nested bounds for the perron root of a nonnegative matrix abstract it is well known that for a nonnegative matrix a the smallest row sum r a and the largest row sum r a provide lower and upper bounds respectively for the perron root of a these bounds are generalized for a partitioned nonnegative matrix a the new bounds are better than r a and r a and they can be further improved by a refinement of the partition known monotonicity and convergence properties of r a and r a are generalized for the new bounds

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

on the rate of convergence of the alternating projection method in finite dimensional spaces abstract using the results of smith solmon and wagner k smith d solomon s wagner practical and mathematical aspects of the problem of reconstructing objects from radiographs bull amer math soc 83 1977 1227 1270 and nelson and neumann s nelson m neumann generalizations of the projection method with application to sor theory for hermitian positive semidefinite linear systems numer math 51 1987 123 141 we derive new estimates for the speed of the alternating projection method and its relaxed version in r m these estimates can be computed in at most o m 3 arithmetic operations unlike the estimates in papers mentioned above that require spectral information the new and old estimates are equivalent in many practical cases in cases when the new estimates are weaker the numerical testing indicates that they approximate the original bounds in papers mentioned above quite well

synchronization reachable topology and synchronization of discrete time linear multi agent systems in this note we consider the synchronization problem for discrete time linear multi agent systems with time varying network topology to describe a new joint connectivity condition for synchronization the concept of synchronization reachable topology is introduced under the conditions that agent dynamics is neutrally stable and network topology is synchronization reachable a synchronization and an exponential synchronization result are obtained using paracontraction theory the two results generalize in a natural way the classical results for the consensus of first order discrete time multi agent systems finally the effectiveness of the results is illustrated by a simulation example

cocyclic hadamard matrices and hadamard groups are equivalent abstract in this paper we prove that the concepts of cocyclic hadamard matrix and hadamard group are equivalent a general procedure for constructing hadamard groups and classifying such groups on the basis of isomorphism type is given to illustrate the ideas cocyclic hadamard matrices over dihedral groups are constructed and the corresponding hadamard groups classified

on some properties of contracting matrices abstract the concepts of paracontracting pseudocontracting and nonexpanding operators have been shown to be useful in proving convergence of asynchronous or parallel iteration algorithms the purpose of this paper is to give characterizations of these operators when they are linear and finite dimensional first we show that pseudocontractivity of stochastic matrices with respect to is equivalent to the scrambling property a concept first introduced in the study of inhomogeneous markov chains this unifies results obtained independently using different approaches secondly we generalize the concept of pseudocontractivity to set contractivity which is a useful generalization with respect to the euclidean norm in particular we demonstrate non hermitian matrices that are set contractive for 2 but not pseudocontractive for 2 or for constant row sum matrices we characterize set contractivity using matrix norms and matrix graphs furthermore we prove convergence results in compositions of set contractive operators and illustrate the differences between set contractivity in different norms finally we give an application to the global synchronization in coupled map lattices

on the ring of 13 13 generic matrices abstract let f be a field of characteristic 0 we investigate the rationality of the center c n of the division algebra of two n n generic matrices over f for n 13 if n is a prime power then the only cases for which c n is known to be stably rational over f are n 2 3 4 5 and 7 with rationality proven for 2 3 and 4 there is a certain z s n lattice denoted a which has played an essential role in the proofs of these results formanek proved the case n 4 by showing that c n is stably isomorphic to f a s n the invariants of f a under the action of s n here a is a z and z is the sign representation of s n lebruyn and bessenrodt proved the cases n 5 and 7 by showing that c n is stably isomorphic to f a s n we show that for n 13 there exists a z s n lattice m which is stably permutation when restricted to the alternating group such that f a m s n and c n are stably isomorphic we then show that a field extension of degree 2 of c n is stably isomorphic to a field extension of degree 2 of a rational extension of f

convergence of sequential and asynchronous nonlinear paracontractions we establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces applications to the solution of linear systems of equations with convex constraints are outlined a first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented

induction theorems on the stable rationality of the center of the ring of generic matrices following procesi and formanek the center of the division ring of rn x n generic matrices over the complex numbers c is stably equivalent to the fixed field under the action of sn of the function field of the group algebra of a zsn lattice denoted by gn we study the question of the stable rationality of the center cn over the complex numbers when rn is a prime in this module theoretic setting let n be the normalizer of an rn sylow subgroup of sn let m be a zsn lattice we show that under certain conditions on m inductionrestriction from n to sn does not affect the stable type of the corresponding field in particular c gn and c zg0zn gn are stably isomorphic and the isomorphism preserves the sn action we further reduce the problem to the study of the localization of gn at the prime n all other primes behave well we also present new simple proofs for the stable rationality of cn over c in the cases r 5 and rn 7

generalizations of the projection methods with application to sor theory for hermitian positive semidefinite linear systems ann n complex matrixb is calledparacontracting if b 2 1 and 0 x n i b bx 2 x 2 we show that a productb b k b k 1 b1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofb in terms of the subdominant singular values of theb i s and in terms of the angles between certain subspaces our results here extend earlier results due to halperin and due to smith solomon and wagner we also determine necessary and sufficient conditions forn numbers in the interval 0 1 to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product in the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the sor iteration matrix associated with ann n hermitian positive semidefinite matrixa none of whose diagonal entries vanish

further results on paracontracting matrices and correction to optimal semistable control in ad hoc network systems a sequential two stage approach this technical note has twofold aspects the first aspect is to present some new results on characterization of matrix paracontraction and the second aspect is to correct some errors in the author s previous publication by means of these newly developed results for paracontraction specifically new necessary and or sufficient conditions to characterize paracontracting matrices using discrete time semistability singular value decomposition normal matrices and range symmetric matrices are provided using these results one can rectify an error in the author s previous publication

convergence properties of art and sor algorithms art algorithms with relaxation parameters are studied for general consistent or inconsistent linear algebraic systemsrx f and a general convergence theorem is formulated the advantage of severe underrelaxation is reexamined and clarified the relationship to solutions obtained by applying sor methods to the equationrr t y f is investigated

on the perturbation of an l2 orthogonal projection abstract the l 2 orthogonal projection onto a subspace is an important mathematical tool which has been widely applied in many fields such as linear least squares problems eigenvalue problems ill posed problems and randomized algorithms in some numerical applications the entries of a matrix will seldom be known exactly so it is necessary to develop some bounds to characterize the effects of the uncertainties caused by matrix perturbation in this paper we establish new perturbation bounds for the l 2 orthogonal projection onto the column space of a matrix which involve upper lower bounds and combined upper lower bounds the new bounds contain some sharper counterparts of the existing ones numerical examples are also given to illustrate our theoretical results

monomial actions of the symmetric group abstract let f be a field let g be a finite group and let m be a g faithful zg lattice we investigate the stable rationality of f m g over f when g is s p the symmetric group on p letters and p is a prime it follows from work of endo miyata lenstra and swan that g faithful zg lattices are in the same flasque class if and only if they have g isomorphic corresponding fields thus the study of flasque classes of zs p lattices plays a fundamental role in this investigation let n be the normalizer of a p sylow subgroup of s p we show that there are classes of zs p lattices for which induction restriction from n to s p does not affect the flasque class in particular we study the flasque class of a specific lattice g p which has the property that f g p s p is stably isomorphic to the center of the division ring of generic matrices formanek linear and multilinear algebra 7 1979 203 212 for a finite group g lattices in the same genus are not in general in the same flasque class however they are for g s n we show that there is a class of zs p lattices containing the genus of g p whose elements are in the flasque of g p

stable rationality of certain pgln quotients stable rationality of the field of matrix invariants m n m n pgl n is proved forn 5 andn 7 in combination with existing results this shows that v pgl n is stably rational wheneverv is an almost free representation ofpgl n andn divides 420 22 3 5 7

the structured h2 optimization problem in this note the structured h 2 optimization problem is defined and two methods are outlined for its solution the first method involves a vectorization kronecker product technique and reduces the optimization problem to two nested projections to circumvent the dimension inflation implicit in this method an iterative algorithm is proposed as an alternative this is an application of von neumann s alternating projection algorithm in hilbert space and can be shown to be applicable to the present problem under two standard rank assumptions a short version of this paper has appeared in halikias 1993

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on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

accelerating the convergence of the method of alternating projections the powerful von neumann halperin method of alternating projections map is an algorithm for determining the best approximation to any given point in a hilbert space from the intersection of a finite number of subspaces it achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the it individual subspaces which make up the intersection the main practical drawback of this algorithm at least for some applications is that the method is slowly convergent in this paper we consider a general class of iterative methods which includes the map as a special case for such methods we study an accelerated version of this algorithm which was considered earlier by gubin polyak and raik 14 and by gearhart and koshy 13 we show that the accelerated algorithm converges faster than the map in the case of two subspaces but is in general it not faster than the map for more than two subspaces however for a symmetric version of the map the accelerated algorithm always converges faster for any number of subspaces our proof seems to require the use of the spectral theorem for self adjoint mappings

products of orthogonal projections and polar decompositions abstract we characterize the sets x of all products pq and y of all products pqp where p q run over all orthogonal projections and we solve the problems arg min p q p q z for z x or y we also determine the polar decompositions and moore penrose pseudoinverses of elements of x

on the acceleration of kaczmarz s method for inconsistent linear systems abstract let the linear system ax b be rectangular but solvable if a is a large sparse matrix then one possibility to solve the system is to use the iterative method of kaczmarz even in the case when the system is unsolvable this method is applicable if the relaxation parameters are kept fixed in a certain interval to get a meaningful result however one has to apply strong underrelaxation we give an interpretation of the limit in terms of a generalized inverse of a and derive bounds with respect to the least squares solution using a result of buoni and varga we estimate the speed of convergence it is further shown that the eigenvalues of the iteration matrix are nearly real numbers and we propose to use chebyshev acceleration this results in a significant speedup