maximization and minimization of the rank and inertia of the hermitian matrix expression a bx bx with applications abstract we give in this paper a group of closed form formulas for the maximal and minimal ranks and inertias of the linear hermitian matrix function a bx bx with respect to a variable matrix x as applications we derive the extremal ranks and inertias of the matrices x x where x is a solution to the matrix equation axb c and then give necessary and sufficient conditions for the matrix equation axb c to have hermitian definite and re definite solutions in addition we give closed form formulas for the extremal ranks and inertias of the difference x 1 x 2 where x 1 and x 2 are hermitian solutions of two matrix equations a 1 x 1 a 1 c 1 and a 2 x 2 a 2 c 2 and then use the formulas to characterize relations between hermitian solutions of the two equations

the common re nnd and re pd solutions to the matrix equations ax c and xb d abstract in this article we consider common re nnd and re pd solutions of the matrix equations ax c and xb d with respect to x where a b c and d are given matrices we give necessary and sufficient conditions for the existence of common re nnd and re pd solutions to the pair of the matrix equations and derive a representation of the common re nnd and re pd solutions to these two equations when they exist the presented examples show the advantage of the proposed approach

rational and polynomial matrix factorizations via recursive pole zero cancellation we develop a recursive algorithm for obtaining factorizations of the type r r1 r2 where all three matrices are rational and r1 is nonsingular moreover the factors r1 and r2 are such that either the poles of r1 1 and r2 are in a prescribed region of the complex plane or their zeros such factorizations cover the specific cases of coprime factorization inner outer factorization gcd extraction and many more the algorithm works on the state space or generalized state space realization of r and derives in a recursive fashion the corresponding realizations of the factors

explicit solutions to the matrix inverse problem ax b abstract recently wu and cain l wu b cain linear algebra and its applications 236 1996 137 146 gave necessary and sufficient conditions for the matrix equationax b to have a solution with respect toa subject to the condition that the hermitian part ofa is non negative definite in this note we give an easier approach which also provides possible explicit expressions fora further necessary and sufficient conditions are derived when the hermitian part ofa is prescribed to be positive definite thus correcting a result in l wu b cain linear algebra and its applications 236 1996 137 146 theorem 5

time reversal in stochastic processes and the dirac equation we consider the motion of a classical particle in 1 1 dimensional space time four probability distributions govern the trajectory of the particle these give the probability of moving to the left or right in space while moving backwards or forwards in time if these probabilities are randomly distributed and if the probability of moving backwards in time is related to the probability of moving forwards in time in a prescribed manner then the master equations for these probabilities give rise to the dirac equation without recourse to direct analytic continuation in contrast when a particle always moves forward in time an analytic continuation is required to recover the dirac equation

an improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects in this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil and implicitly its kronecker structure the algorithm compares favorably to existing ones in terms of elegance versatility and complexity in particular the algorithm without any modification yields the structural invariants associated with a generalized state space system and its system pencil two related geometric aspects are also discussed we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form we extend the notion of deflating subspace to the singular pencil case c 1997 elsevier science b v

re nnd generalized inverses abstract in this paper we give some necessary and sufficient conditions for the existence of re nnd and nonnegative definite 1 3 and 1 4 inverses of a matrix a c n n and completely described these sets moreover we prove that the existence of nonnegative definite 1 3 inverse of a matrix a is equivalent with the existence of its nonnegative definite 1 2 3 inverse and present the necessary and sufficient conditions for the existence of re nnd 1 3 4 inverse of a

numerical solution to a descriptor discrete time algebraic riccati equation abstract we investigate the numerical solution of a special type of descriptor discrete time riccati equation involved in the solution of several key problems in robust control formulated under very general hypotheses necessary and sufficient existence conditions together with computable formulas are given for the stabilizing and antistabilizing solutions in terms of an associated matrix pencil two particular applications and numerical experiments of our results are included for squaring down with l norm preservation and j j spectral factorization both for a completely general discrete time system

mathcal h _ infty control problem for discrete time algebraic dynamical systems this paper considers the suboptimal mathcal h infty control problem for linear discrete time algebraic dynamical systems such systems can be formally described as linear and time invariant discrete time systems whose transfer function matrix is allowed to be improper or polynomial the parametrization of output feedback controllers is given in a realization based setting involving two generalized algebraic riccati equations and features the same elegant simplicity of the standard proper case two relevant numerical examples prove the effectiveness of our approach

equations ax c and xb d in rings and rings with involution with applications to hilbert space operators abstract this paper reviews the equations ax c and xb d from a new perspective by studying them in the setting of associative rings with or without involution results for rectangular matrices and operators between different banach and hilbert spaces are obtained by embedding the rectangles into rings of square matrices or rings of operators acting on the same space necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian skew hermitian reflexive antireflexive positive and real positive solutions and the general solutions are described in terms of the original elements or operators new results are obtained and many results existing in the literature are recovered and corrected

constructive solutions to spectral and inner outer factorizations with respect to the disk we use state space realizations to solve the spectral and inner outer factorization problems with respect to the unit disk formulated for a completely general rational matrix the algorithm is based on orthogonal transformations and standard reliable procedures for solving stein and riccati equations the formulas apply in particular to general descriptor systems or polynomial matrices the main novelty is that we allow for arbitrary rank poles and zeros on the unit circle or at infinity

minimal indices cancellation and rank revealing factorizations for rational matrix functions we study the problem of eliminating the minimal indices the indices of a minimal polynomial basis of the null space of a rational matrix function by multiplication with a suitably chosen invertible rational matrix function we derive the class of all invertible factors that dislocate the minimal indices to certain zero locations and feature minimal mcmillan degree we impose additional conditions on the factor like being j unitary or j inner either with respect to the imaginary axis or to the unit circle and characterize the classes of solutions en route we extend the well known rank revealing factorization of a constant matrix to rational matrix functions the results are completely general and apply in particular to matrices which are polynomial strictly proper or improper rank deficient with arbitrary poles and zeros including at infinity all characterization are made by using descriptor realizations while the associated computations are performed by employing constant unitary transformations and standard reliable procedures for eigenvalue assignment or for solving riccati equations

j factorizations of a general discrete time system abstract the problems of extending the j lossless conjugation the j j spectral factorization and the j j lossless factorization to a completely general discrete time system are considered existence conditions together with a numerically sound prototype algorithm for computing the factors are provided for a system having any type of singularity including arbitrary normal rank poles and zeros at infinity at zero or on the unit circle

zero cancellation for general rational matrix functions the problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints firstly the class of all factors that dislocate the zeros and feature minimal mcmillan degree are derived further necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being j unitary or j inner either with respect to the imaginary axis or to the unit circle the main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial proper or improper rank deficient with arbitrary poles and zeros including at infinity a particular attention is paid to the numerically sound construction of solutions by employing at each stage unitary transformations reliable numerical algorithms for eigenvalue assignment and efficient lyapunov equation solvers

youla like parametrizations subject to qi subspace constraints consider that a linear time invariant lti plant is given and that we wish to design a stabilizing controller for it admissible controllers are lti and must belong to a pre selected subspace that may impose structural restrictions such as sparsity constraints the subspace is assumed to be quadratically invariant qi with respect to the plant which from prior results guarantees that there is a convex parametrization of all admissible stabilizing controllers provided that an initial admissible stable stabilizing controller is provided this paper addresses the previously unsolved problem of extending youla s classical parametrization so that it admits qi subspace constraints on the controller in contrast with prior parametrizations the one proposed here does not require initialization and it does not require the existence of a stable stabilizing controller the main idea is to cast the stabilizability constraint as an exact model matching problem with stability restrictions which can be tackled using existing methods furthermore we show that when it exists the solution of the exact model matching problem can be used to compute an admissible stabilizing controller applications of the proposed parametrization on the design of norm optimal controllers via convex methods are also explored an illustrative example is provided and a special case is discussed for which the exact model matching problem has a unique and easily computable solution

j j lossless factorization based on conjugation this paper is concerned with a derivation of the state space form of the j j lossless factorization which contains both the inner outer factorization and the spectral factorization of positive matrices as special cases also the j j lossless factorization gives a unified framework of h control theory we use the method of conjugation which makes the derivation much simpler than the previous literature most of which used the technique of j j spectral factorization a necessary and sufficient condition is represented in terms of two riccati equations one of which is degenerated

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

a continuous time generalization of the persistent random walk abstract we develop the formalism for a continuous time generalization of the persistent random walk by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject this generalization leads to evolution equations in the time domain that differ and are of higher order than the telegrapher s equation which is found in the case of the markovian persistent random walk

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

computation of the general j j prime lossless factorization the j j lossless factorization problem for a generalized state space system whose transfer matrix is completely general is investigated the main novelty is that we remove all restrictive hypotheses and allow for arbitrary normal rank poles and zeros on the imaginary axis or at infinity the numerically sound formulas for the solution bear the same nice expressions and striking simplicity of the inner outer factorization and may be equally well applied for the j j spectral factorization

an expression of the general common least squares solution to a pair of matrix equations with applications in this paper an explicit representation of the general common least squares solution to the pair of matrix equations a 1xb 1 c 1 and a 2xb 2 c 2 is obtained furthermore we use this result to determine the condition for the existence of a hermitian least squares solution to the matrix equation axb c and the expression of the general hermitian least squares solution is also given special attention is paid to consider the existence of hermitian 1 i inverses of a i 3 4 and the representations of the hermitian generalized inverses are presented finally a numerical example is given to illustrate our result

computation of general inner outer and spectral factorizations in this paper we solve two problems in linear systems theory the computation of the inner outer and spectral factorizations of a continuous time system considered in the most general setting we show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic riccati equation of order usually much smaller than the mcmillan degree of the transfer function matrix of the system the proposed procedures are completely general being applicable for a polynomial proper improper system whose transfer function matrix could be rank deficient and could have poles zeros on the imaginary axis or at infinity as an application we discuss the extension to the case of rational matrices of the complete orthogonal decomposition of a constant matrix numerical refinements and examples illustrating the proposed approach are discussed in detail

the re positive definite solutions to the matrix inverse problem ax b abstract an n n complex matrix a is termed re positive definite re pd if the real part of x ax is positive for every nonzero complex n vector x this paper is concerned with constructing complex re pd matrices and solving the following matrix inverse problem given complex matrices x and b find the set of all complex re pd matrices a such that ax b

individual based and mean field modeling of direct aggregation we introduce two models of biological aggregation based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood in the first order model the location of each individual is subject to a density dependent random walk while in the second order model the density dependent random walk acts on the velocity variable together with a density dependent damping term the main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals instead aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density we formally derive the corresponding mean field limits leading to nonlocal degenerate diffusions then we carry out the mathematical analysis of the first order model in particular we prove the existence of weak solutions and show that it allows for measure valued steady states we also perform linear stability analysis and identify conditions for pattern formation moreover we discuss the role of the nonlocality for well posedness of the first order model finally we present results of numerical simulations for both the first and second order model on the individual based and continuum levels of description