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

decomposition of matrices into three involutions abstract in this paper we generalize some results of c s ballantine concerning products of three n n complex involutions we prove that each n n complex matrix a with determinant 1 and dimker a n 2 for all c is the product of three involutions on the other hand we show that if an n n complex matrix a is the product of three involutions then m 2n r 3 and m 3n 4 where m dimker a and r dimker a 3 for any 0 and 4 1 we also completely characterize products of three 5 5 complex involutions

ϕs orthogonal matrices abstract let s m n r be such that s 2 i or s 2 i for a m n c define s a s 1 a t s we study s orthogonal matrices those a m n c that satisfy s a a 1 let f r or f c we show that every s orthogonal a m n f has a polar decomposition a pu with p u m n f p is positive definite u is unitary and both factors are s orthogonal we show that if a is s orthogonal and normal and if 1 is not an eigenvalue of a then there exists a normal s skew symmetric n that is s n n such that a e in we also take a look at the particular cases s h k 0 i k i k 0 and s l k i k i n k

on the parametrization of all decentralized stabilizing controllers abstract a novel parametrization of all decentralized stabilizing controllers is provided in terms of a single parameter q that is constrained to satisfy a finite number of quadratic equations the results are presented in a general algebraic framework and apply to linear time invariant systems implications of this parametrization for decentralized control system performance are also discussed

decomposition of matrices into 2 involutions abstract we consider the group slnf of all n by n matrices with determinant 1 over a field let res a denote rank a i for any a in slnf a matrix a is called a k involution if a2 i and res a k we prove that every matrix a in slnf is the product of at most res a 1 2 2 2 involutions for n 3 if char f 2 and for n 4 if char f 2

state space solutions to hadamard weighted ℋ and ℋ 2 control problems variations of the standard h and h 2 control problems are presented state space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while i satisfying an h frobenius norm bound or ii minimizing the h 2 norm of the closed loop the optimal h frobenius case is analysed and it is shown that the optimal controller is often unique in contrast to the standard h control problem the solutions of an algebraic riccati equation are examined in the h 2 case to reveal the structure of the h 2 optimal controller

sums and products of quasi nilpotent operators it is proved that a bounded operator on hilbert space is the sum of two quasi nilpotent operators if and only if it is not a non zero scalar plus a compact operator necessary conditions and sufficient conditions for an operator to be the product of two quasi nilpotent operators are given

on the ϕj polar decomposition of matrices abstract we present new results on the j polar decomposition of matrices we show that every symplectic matrix may be written as a product of symplectic operation matrices we present a simple form attained under symplectic equivalence which makes it easy to determine if a matrix does not have a j polar decomposition we also determine the rank 4 matrices with j polar decomposition

symplectic spaces and pairs of symmetric and nonsingular skew symmetric matrices under congruence abstract let f be a field of characteristic not 2 and let a b be a pair of n n matrices over f in which a is symmetric and b is skew symmetric a canonical form of a b with respect to congruence transformations s t a s s t b s was given by sergeichuk 1988 25 up to classification of symmetric and hermitian forms over finite extensions of f we obtain a simpler canonical form of a b if b is nonsingular such a pair a b defines a quadratic form on a symplectic space that is on a vector space with scalar product given by a nonsingular skew symmetric form as an application we obtain known canonical matrices of quadratic forms and hamiltonian operators on real and complex symplectic spaces

products of positive semi definite matrices abstract it is known that every complex square matrix with nonnegative determinant is the product of positive semi definite matrices there are characterizations of matrices that require two or five positive semi definite matrices in the product however the characterizations of matrices that require three or four positive semi definite matrices in the product are lacking in this paper we give a complete characterization of these two types of matrices with these results we give an algorithm to determine whether a square matrix can be expressed as the product of k positive semi definite matrices but not fewer for k 1 2 3 4 5

powers of 1 cyclic conjugacy classes in the groups gln f and sln f abstract let a gl n f where f is a field we say that a is 1 cyclic if a is similar to a matrix of the form a diag a 1 a 2 a k where a i gl l i f is cyclic for 1 i k l 1 0 1 and l i 2 for 2 i k it is shown that if a gl n f is 1 cyclic where n 2 and f 4 then every nonscalar matrix m gl n f whose determinant equals det a 4 is the product of four matrices which are similar to a under matrices of sl n f the problem of expressing a scalar matrix as a product of similar 1 cyclic matrices is also discussed the above result is applied to problems of factorizing matrices in the group sl n f into products of unipotent matrices of index 2 and into products of matrices of fixed finite order

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

heavy traffic analysis of a controlled multiclass queueing network via weak convergence methods the workload formulation due to harrison and coworkers of multiclass queueing networks has been fundamental to its analysis until recently there was no actual theory which started with the physical queue and showed that under heavy traffic conditions the optimal costs could be approximated by those for an optimization problem using the limit workload equations recently this was done via viscosity solution methods by martins shreve and soner for one important class for this same class of problems and including the cases not treated there we use weak convergence methods to show that the sequence of optimal costs for the original network converges to the optimal cost for the workload limit problem the proof is simpler and allows weaker and non markovian conditions it uses current techniques in weak convergence analysis it seems to be the first analysis of such multiclass workload problems by weak convergence methods the general structure of the development seems applicable to the analysis of more complex systems

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

every 2n by 2n complex matrix is a sum of three symplectic matrices abstract let j 2 n 0 i n i n 0 an a m 2 n c is called symplectic if a t j 2 n a j 2 n if n 1 then we show that every matrix in m 2 n c is a sum of two symplectic matrices if n 1 then we show that every matrix in m 2 n c is a sum of three symplectic matrices moreover we show that some matrices cannot be written with less than three symplectic matrices we also show that for every a m 2 n c there exist symplectic p q m 2 n c and b c d m n c such that p a q b c 0 d if a is skew hamiltonian j 2 n 1 a t j 2 n a then we show that a is a sum of two symplectic matrices

products of nilpotent matrices we show that any complex singular square matrix t is a product of two nilpotent matrices a and b with rank a rank b rank t except when t is a 2 x 2 nilpotent matrix of rank one an n x n complex matrix t is nilpotent if t 0 it is easily seen that a product of finitely many nilpotent matrices must be singular the purpose of this note is to prove the converse

pathwise optimization for optimal stopping problems we introduce the pathwise optimization po method a new convex optimization procedure to produce upper and lower bounds on the optimal value the price of a high dimensional optimal stopping problem the po method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales which we dub the martingale duality approach we demonstrate via numerical experiments that the po method produces upper bounds of a quality comparable with state of the art approaches but in a fraction of the time required for those approaches as a by product it yields lower bounds and suboptimal exercise policies that are substantially superior to those produced by state of the art methods the po method thus constitutes a practical and desirable approach to high dimensional pricing problems furthermore we develop an approximation theory relevant to martingale duality approaches in general and the po method in particular our analysis provides a guarantee on the quality of upper bounds resulting from these approaches and identifies three key determinants of their performance the quality of an input value function approximation the square root of the effective time horizon of the problem and a certain spectral measure of predictability of the underlying markov chain as a corollary to this analysis we develop approximation guarantees specific to the po method finally we view the po method and several approximate dynamic programming methods for high dimensional pricing problems through a common lens and in doing so show that the po method dominates those alternatives this paper was accepted by wei xiong stochastic models and simulation

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

on some factorizations of operators abstract given two subsets a and b of the algebra of bounded linear operators on a hilbert space h we denote by a b ab a a b b the goal of this article is to describe a b if a and b denote classes of projections partial isometries positive semidefinite operators etc moreover fixed t a b we shall describe a b t a b a b ab t p 1 a b t a a t ab for some b b and p 2 a b t b b t ab for some a a

hamiltonian square roots of skew hamiltonian matrices abstract we present a constructive existence proof that every real skew hamiltonian matrix w has a real hamiltonian square root the key step in this construction shows how one may bring any such w into a real quasi jordan canonical form via symplectic similarity we show further that every w has infinitely many real hamiltonian square roots and give a lower bound on the dimension of the set of all such square roots some extensions to complex matrices are also presented

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

the length problem for a sum of idempotents abstract it was shown by r e hartwig and m s putcha and independently by p y wu that the complex square matrix t is a sum of finitely many idempotent matrices if and only if tr t is an integer and tr t rank t we determine the minimal number of required idempotents for such matrices in terms of their traces and sizes and we completely solve the problem for matrices t with size 2 3 4 or 5 also we give some sufficient necessary conditions for a complex matrix t to be the sum of three or more idempotents

products of cyclic similarity classes in the groups gln f abstract let g gln f be the group of all n n invertible matrices over a field f where f 4 and n 3 let j1 j2 be two prescribed cyclic elements of g in jordan canonical form and assume that all the eigenvalues of j1 or j2 lie in f it is shown that any nonscalar matrix in g can be written as a product of two matrices having the prescribed jordan canonical forms j1 j2 subject only to the obvious 0 determinant condition as a corollary we show that if f 4 the group psln f has a conjugacy class c such that c2 psln f another corollary states that every nonscalar matrix in g can be written as a product of two cyclic unipotent matrices

stochastic dominance constrained markov decision processes we are interested in risk constraints for infinite horizon discrete time markov decision processes mdps starting with average reward mdps we show that increasing concave stochastic dominance constraints on the empirical distribution of reward lead to linear constraints on occupation measures an optimal policy for the resulting class of dominance constrained mdps is obtained by solving a linear program we compute the dual of this linear program to obtain average dynamic programming optimality equations that reflect the dominance constraint in particular a new pricing term appears in the optimality equations corresponding to the dominance constraint we show that many types of stochastic orders can be used in place of the increasing concave stochastic order we also carry out a parallel development for discounted reward mdps with stochastic dominance constraints a portfolio optimization example is used to motivate the paper

approximate dynamic programming via a smoothed linear program we present a novel linear program for the approximation of the dynamic programming cost to go function in high dimensional stochastic control problems lp approaches to approximate dp have typically relied on a natural projection of a well studied linear program for exact dynamic programming such programs restrict attention to approximations that are lower bounds to the optimal cost to go function our program the smoothed approximate linear program is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable doing so appears to have several advantages first we demonstrate bounds on the quality of approximation to the optimal cost to go function afforded by our approach these bounds are in general no worse than those available for extant lp approaches and for specific problem instances can be shown to be arbitrarily stronger second experiments with our approach on a pair of challenging problems the game of tetris and a queueing network control problem show that the approach outperforms the existing lp approach which has previously been shown to be competitive with several adp algorithms by a substantial margin

dynamic bid prices in revenue management we formally derive the standard deterministic linear program lp for bid price control by making an affine functional approximation to the optimal dynamic programming value function this affine functional approximation gives rise to a new lp that yields tighter bounds than the standard lp whereas the standard lp computes static bid prices our lp computes a time trajectory of bid prices we show that there exist dynamic bid prices optimal for the lp that are individually monotone with respect to time we provide a column generation procedure for solving the lp within a desired optimality tolerance and present numerical results on computational and economic performance

a dynamic traveling salesman problem with stochastic arc costs we propose a dynamic traveling salesman problem tsp with stochastic arc costs motivated by applications such as dynamic vehicle routing in which the cost of a decision is known only probabilistically beforehand but is revealed dynamically before the decision is executed we formulate this as a dynamic program dp and compare it to static counterparts to demonstrate the advantage of the dynamic paradigm over an a priori approach we then apply approximate linear programming alp to overcome the dp s curse of dimensionality obtain a semi infinite linear programming lower bound and discuss its tractability we also analyze a rollout version of the price directed policy implied by our alp and derive worst case guarantees for its performance our computational study demonstrates the quality of a heuristically modified rollout policy using a computationally effective a posteriori bound

multiplicative maps on matrices that preserve the spectrum abstract let m n denote the set of all n n matrices over the complex numbers n 2 let a n m n be either the set of all invertible matrices the set of all unitary matrices or a multiplicative semigroup containing the singular matrices theorem if a n m n is a spectrum preserving multiplicative map then there exists a matrix r in m n such that s r 1 sr for all s in a n

factorizations of upper triangular matrices if d is an integral domain in which every element can be factored as a product of irreducible elements then every element a tn d the semigroup of upper triangular matrices with nonzero determinant can be factored as a product of irreducible elements we classify both the irreducible elements and units of this cancellative semigroup having achieved this we provide a means of measuring the uniqueness or non uniqueness of these factorizations in terms of the uniqueness or non uniqueness of factorizations in d in addition to using well studied invariants from factorization theory of commutative semigroups we introduce a new tool weak transfer homomorphisms which are more appropriate for noncommutative settings

products of cyclic conjugacy classes in the groups psl n f abstract let g be the group psl n f where f is a field and n 3 if c is a conjugacy class of g denote c 1 c 1 c c the following is proved 1 if c1 c2 c3 are cyclic conjugacy classes of g then c1c2c3 g 1g 2 if f is algebraically closed and c1 c2 are cyclic conjugacy classes of g then c1c2 g if and only if c1 c 12 generalizations of these results concerning factorizations of a given nonscalar invertible matrix as a product of two or three cyclic matrices each lying in a prescribed conjugacy class of gl n f or sl n f are discussed

normal matrices an update abstract a list of seventy conditions on an n n complex matrix a equivalent to its being normal published nearly ten years ago by grone johnson sa and wolkowicz has proved to be very useful hoping that in an extended form it will be even more helpful we compile here another list of about twenty conditions they either have been overlooked by the authors of the original list or have appeared during the last decade

heavy traffic convergence of a controlled multiclass queueing system this paper provides a rigorous proof of the connection between the optimal sequencing problem for a two station two customer class queueing network and the problem of control of a multidimensional diffusion process obtained as a heavy traffic limit of the queueing problem in particular the diffusion problem which is one of singular control of a brownian motion is used to develop policies which are shown to be asymptotically nearly optimal as the traffic intensity approaches one in the queueing network the results are proved by a viscosity solution analysis of the related hamilton jacobi bellman equations

factorizations of operator matrices abstract this note gives explicit factorizations of a 2 2 operator matrix as a product of an upper triangular operator matrix and an involutory unitary or j unitary operator matrix a pattern is given for construction of factorizations of this kind

each symplectic matrix is a product of four symplectic involutions abstract gustafson halmos and radjavi in 1973 proved that each matrix a with det a 1 is a product of four involutions we prove that these involutions can be taken to be symplectic if a is symplectic every symplectic matrix has unit determinant using this result we give an alternative proof of laffey s theorem that every nonsingular even size matrix is a product of skew symmetric matrices ballantine in 1978 proved that each matrix a with det a 1 is a product of four coninvolutions we prove that these coninvolutions can be taken to be symplectic if a is symplectic we also prove that each hamiltonian matrix is a sum of two square zero hamiltonian matrices

every real symplectic matrix is a product of real symplectic involutions abstract a 2n by 2n matrix a is symplectic if a t 0 i i 0 a 0 i i 0 it is known that if n 1 then every 2n by 2n complex symplectic matrix is a product of four symplectic involutions we consider the real case we give an example of a real symplectic matrix which is a product of two complex symplectic involutions but is not a product of two real symplectic involutions we show that every 4 by 4 real symplectic matrix is a product of four real symplectic involutions we use this result to show that for n 1 every 2n by 2n real symplectic matrix is a product of a finite number of real symplectic involutions

decomposition of a scalar operator into a product of unitary operators with two points in spectrum abstract we consider products of unitary operators with at most two points in their spectra 1 and e i we prove that the scalar operator e i i is a product of k such operators if 1 1 k 3 k 1 1 k 3 for k 5 also we prove that for e i 1 only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class as a corollary we show that every unitary operator on an infinite dimensional space is a product of finitely many such operators

the product of matrix subspaces abstract in factoring matrices into the product of two matrices operations are typically performed with elements restricted to matrix subspaces such modest structural assumptions are realistic for example in large scale computations this paper is concerned with analyzing associated matrix geometries curvature of the product of two matrix subspaces is assessed the case of vanishing curvature gives rise to the notion of factorizable matrix subspace this can be regarded as an analogue of the internal zappa szep product in group theory interpreted in this way several classical instances are encompassed by this structure the craig sakamoto theorem fits naturally into this framework

generalized polynomial approximations in markovian decision processes abstract fitting the value function in a markovian decision process by a linear superposition of m basis functions reduces the problem dimensionality from the number of states down to m with good accuracy retained if the value function is a smooth function of its argument the state vector this paper provides for both the discounted and undiscounted cases three algorithms for computing the coefficients in the linear superposition linear programming policy iteration and least squares

a cost shaping linear program for average cost approximate dynamic programming with performance guarantees we introduce a new algorithm based on linear programming for optimization of average cost markov decision processes mdps the algorithm approximates the differential cost function of a perturbed mdp via a linear combination of basis functions we establish a bound on the performance of the resulting policy that scales gracefully with the number of states without imposing the strong lyapunov condition required by its counterpart in de farias and van roy de farias d p b van roy 2003 the linear programming approach to approximate dynamic programming oper res 51 6 850 865 we investigate implications of this result in the context of a queueing control problem

using lagrangian relaxation to compute capacity dependent bid prices in network revenue management we propose a new method to compute bid prices in network revenue management problems the novel aspect of our method is that it explicitly considers the temporal dynamics of the arrivals of the itinerary requests and generates bid prices that depend on the remaining leg capacities our method is based on relaxing certain constraints that link the decisions for different flight legs by associating lagrange multipliers with them in this case the network revenue management problem decomposes by the flight legs and we can concentrate on one flight leg at a time when compared with the so called deterministic linear program we show that our method provides a tighter upper bound on the optimal objective value of the network revenue management problem computational experiments indicate that the bid prices obtained by our method perform significantly better than the ones obtained by standard benchmark methods

pairs of alternating forms and products of two skew symmetric matrices let k be a field and let s t s1 t1 be skew symmetric matrices over k with s s1 both nonsingular if k has characteristic 2 a skew symmetric matrix is a symmetric one with zero diagonal it is shown that there exists a nonsingular matrix p over k with p sp s1 p tp t1 where p denotes the transpose of p if and only if s 1t and s 11t1 are similar it is also shown that a 2m 2m matrix over k can be factored as st with s t skew symmetric and s nonsingular if and only if a is similar to a matrix direct sum b b where b is an m m matrix over k this is equivalent to saying that all elementary divisors of a occur with even multiplicity an extension of this result giving necessary and sufficient conditions for a square matrix to be so expressible without assuming that either s or t is nonsingular is included

products of symplectic normal matrices abstract a matrix a m 2 n c is symplectic if a t 0 i n i n 0 a 0 i n i n 0 we show that every symplectic matrix is a product of a symplectic unitary and a symplectic skew hermitian matrix we show that every symplectic matrix is a product of four symplectic skew hermitian matrices or a product of four symplectic hermitian matrices we give the possible jordan canonical forms of symplectic matrices which can be written as a product of a symplectic hermitian and a matrix which is either symplectic hermitian or symplectic skew hermitian

on the diagonalizability of a matrix by a symplectic equivalence similarity or congruence transformation abstract a symplectic matrix s c 2 n 2 n satisfies s j 1 s t j for j 0 i n i n 0 r 2 n 2 n we will consider symplectic equivalence similarity and congruence transformations and answer the question under which conditions a 2 n 2 n matrix is diagonalizable under one of these transformations in particular we will give symplectic analogues of the singular value decomposition and the takagi factorization

achieving diagonal dominance there are many occasions when one would like to see whether a transfer function can be made diagonally dominant there is the obvious and important case of system design using rosenbrock s methods but one also wants to use diagonal dominance in a different manner in robustness studies and in several nonlinear system analysis techniques commonly the compensators or similarity transforms used to achieve dominance are designed in an ad hoc way this paper shows how in the particular case of diagonal compensators or similarity transforms there is a very easy way based on perron frobenius theory of obtaining optimal reduction in the off diagonal elements

incremental constraint projection methods for variational inequalities we consider the solution of strongly monotone variational inequalities of the form f x x x ge 0 f x x x 0 for all x in x x x we focus on special structures that lend themselves to sampling such as when x x is the intersection of a large number of sets and or f f is an expected value or is the sum of a large number of component functions we propose new methods that combine elements of incremental constraint projection and stochastic gradient these methods are suitable for problems involving large scale data as well as problems with certain online or distributed structures we analyze the convergence and the rate of convergence of these methods with various types of sampling schemes and we establish a substantial rate of convergence advantage for random sampling over cyclic sampling

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

products of positive semidefinite matrices abstract we characterize the complex square matrices which are expressible as the product of finitely many positive semidefinite matrices a matrix t can be expressed as such if and only if det t 0 moreover the number of factors can always be limited to five we also determine those matrices which can be expressed as the product of two or four positive semidefinite matrices these results are analogous to the ones obtained before by c s ballantine for products of positive definite matrices

skew coninvolutory matrices we study the properties of skew coninvolutory ee i matrices and derive canonical forms and a singular value decomposition we study the matrix function s a sa 1s 1 defined on nonsingular matrices and with s satisfying ss i or ss i we show that every square nonsingular a may be written as a xy with s x x and s y y 1 we also give necessary and sufficient conditions on when a nonsingular matrix may be written as a product of a coninvolutory matrix and a skew coninvolutory matrix or a product of two skew coninvolutory matrices moreover when a is similar to a 1 or when a is similar to a 1 or when a is similar to a or when a is similar to a we determine the possible jordan canonical forms of a for which the similarity matrix may be taken to be skew coninvolutory

products of diagonalizable matrices abstract let f denote a field such that char f 2 it is shown that every square matrix over f is expressible as a product of two diagonalizable matrices except when f gf 3 in which case three diagonalizable matrices are needed in general partial results for the case where char f 2 is also presented finally the extent to which the nullities of these factors can be prescribed is also investigated

the linear programming approach to approximate dynamic programming the curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of large scale stochastic control problems we study an efficient method based on linear programming for approximating solutions to such problems the approach fits a linear combination of pre selected basis functions to the dynamic programming cost to go function we develop error bounds that offer performance guarantees and also guide the selection of both basis functions and state relevance weights that influence quality of the approximation experimental results in the domain of queueing network control provide empirical support for the methodology

approximate linear programming for average cost mdps we consider the linear programming approach to approximate dynamic programming with an average cost objective and a finite state space using a lagrangian form of the linear program lp the average cost error is shown to be a multiple of the best fit differential cost error this result is analogous to previous error bounds for a discounted cost objective second bounds are derived for average cost error and performance of the policy generated from the lp that involve the mixing time of the markov decision process mdp under this policy or the optimal policy these results improve on a previous performance bound involving mixing times

on a new method for h_ 2 based decomposition this note proposes a simple linear matrix inequality lmi system for the design of static precompensators to reduce the interactions of a multivariable system the approach is based on minimizing the h2 norm of a modified system and the lmis are derived from the well known technique of pseudodiagonalisation the approach is applied to two complex real life benchmark problems with high levels of interaction it is shown that its performance is significantly better than previously proposed lmi optimization techniques for designing static precompensators

a numerical method for solving singular stochastic control problems singular stochastic control has found diverse applications in operations management economics and finance however in all but the simplest of cases singular stochastic control problems cannot be solved analytically in this paper we propose a method for numerically solving a class of singular stochastic control problems we combine finite element methods that numerically solve partial differential equations with a policy update procedure based on the principle of smooth pasting to iteratively solve hamilton jacobi bellman equations associated with the stochastic control problem a key feature of our method is that the presence of singular controls simplifies the procedure we illustrate the method on two examples of singular stochastic control problems one drawn from economics and the other from queueing systems

note on the factorization of a square matrix into two hermitian or symmetric matrices this paper presents elementary proofs of the factorization of a square matrix into two hermitian or symmetric matrices

factorizations of a matrix into symmetric matrices with prescribed nullities abstract necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k 2 symmetric matrices with prescribed nullities a generalization of the case k 2 to hermitian matrices is also presented

products of idempotent linear transformations in 1966 j m howie characterised the transformations of an arbitrary set that can be written as a product under composition of idempotent transformations of the same set in 1967 j a erdos considered the analogous problem for linear transformations of a finite dimensional vector space and in 1983 r j dawlings investigated the corresponding idea for bounded operators on a separable hilbert space in this paper we study the case of arbitrary vector spaces

a new method for singular value loop shaping in design of multiple channel controllers in this note simple symmetric interval bounds on the singular values of a matrix based on its gershgorin disks are proposed this allows the gershgorin theorem to be used not only to provide information about the location of the eigenvalues of a matrix but also its singular values this is utilized for the proposition of a new design technique for singular value loop shaping based on the diagonal dominance methodology for design of linear multivariable plants in return this allows multiple channel simply structured controllers to be designed with a view to robustness and to meet constraints and specifications on the behavior of its singular values a design example is given demonstrating the effectiveness of this approach