invertibility of irreducible matrices abstract we obtain new sufficient conditions for invertibility of an irreducible complex matrix remarks are also given on eigenvalues and the associated eigenvectors that lie on the boundary of various spectrum inclusion regions of an irreducible matrix our results extend strengthen or clarify the recent work of brualdi brualdi and mellendorff farid solov ev and zhang and gu

maximal minimal resolutions that is exact everywhere except at f0 where the cokernel is m such a resolution always exists if r is a noetherian n graded ring with r0 a field and m is a finitely generated z graded r module then a resolution may be constructed in a minimal way in which each fi is graded and each differential is homogeneous of degree 0 the rank of the ith free module in a minimal free resolution of m is an invariant of m called the ith betti number of m and denoted i m likewise r ij m the number of elements of degree j in a minimal set of homogeneous generators of fi is also an invariant of m called the i j th graded betti number of m in this paper we study modules with maximal graded betti numbers more precisely consider a set consisting of pairs r m where r is a noetherian n graded ring with r0 a field and m is a finitely generated graded r module is there some r m such that ij m ij m for every i and j and every r m there are trivial examples in which the answer is yes e g has only one element or no e g consists of all such pairs theorems 1 and 2 below give affirmative answers for certain sets defined by conditions on the hilbert series and the depths of r and m theorem 3 allows one to compute the maximal graded betti numbers for the pairs in the sets considered in theorems 1 and 2 throughout this paper k denotes a field and q the polynomial ring k x1 xn with the usual n grading given by degxi 1 the hilbert series of a finitely generated graded q module m is hm s dimk mis we define d lexicographic ideals and submodules in section 2 the poincare series of an r module m is p m s t ij m s t we write 4 for coefficient wise inequality of laurent series with coefficients in z

robust stabilization of a system with delays in control a robust stabilization problem is discussed for a system with delays in control the authors derive a robust stabilizing law against additive perturbations then motivated by the feature of predictive controllers they investigate the structure of robust stabilizing law and provide some interpretations on the relation between the value of time delay and the achievable margin of robust stabilizability

spectral inequalities for matrix exponentials abstract this note generalizes an inequality of bernstein as follows if c is an n n complex matrix and c k is the k th compound of c 1 k n n n k and if the eigenvalues of c k are labeled in order of decreasing magnitude 1 c k 2 c k n c k define the partial trace tr k i c by tr k i c h 1 i h c k i 1 n then for any complex n n matrix a tr k i e a e a tr k i e a a i 1 n with equality if a is normal or k n a spectral inequality of k fan is also generalized through the use of compound matrices

the high road to an exponential formula a conjectured exponential formula is proved using a recent result of klyachko linear algebra appl 319 2000 37 then the exponential formula is applied to confirm the equality of two types of spectral triples used to describe all homogeneous eigenvalue inequalities for sum of hermitian matrices and product of positive definite matrices

universal lex ideal approximations of extended hilbert functions and hamilton numbers abstract let r h denote the polynomial ring in variables x 1 x h over a specified field k we consider all of these rings simultaneously and in each use lexicographic lex monomial order with x 1 x h given a fixed homogeneous ideal i in r h for each d there is unique lex ideal generated in degree at most d whose hilbert function agrees with the hilbert function of i up to degree d when we consider i r n for n h the set b d i n of minimal generators for this lex ideal in degree at most d may change but b d i n is constant for all n 0 we let b d i denote the set of generators one obtains for all n 0 and we let b d b d i be its cardinality the sequences b 1 b d obtained in this way may grow very fast remarkably even when i x 1 2 x 2 2 one obtains a very interesting sequence 0 2 3 4 6 12 924 409620 this sequence is the same as h d 1 1 for d 2 where h d is the dth hamilton number the hamilton numbers were studied by hamilton and by hammond and sylvester because of their occurrence in a counting problem connected with the use of tschirnhaus transformations in manipulating polynomial equations

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

the spectrum of a hermitian matrix sum abstract an announcement by lidskii b v lidskii functional anal appl 10 1982 76 77 russian 139 140 english claimed to establish the explicit description of the spectrum of a hermitian matrix sum in the form conjectured by horn a horn pacific j math 16 1962 225 241 but no supporting proof has been published this paper begins an analysis of the claim and is the first step towards bridging the distance between the elementary methods of horn s 1962 paper and the partial solution of the same problem using noncommutative harmonic analysis by dooley et al i dooley r repka n wildberger lin mult alg 36 1993 79 102 the methods of horn lidskii lead to combinatorial issues of independent interest

galerkin approximation in modeling of controlled distributed parameter flexible systems abstract numerical study of control problems for linear distributed parameter flexible mechanical systems requires finite dimensional modeling of the system we study the conditions for h covergence of the transfer functions of finite dimensional galerkin approximations this convergence ensures that control problems can be solved by using a galerkin model a stabilization problem for the angular position of a bernoulli euler beam illustrates the theoretical consideration the beam with an inertial drive represents a flexible manipulator link

robust stabilization of infinite dimensional systems by finite dimensional controllers abstract the problem of robustly stabilizing an infinite dimensional system with transfer function g subject to an additive perturbation is considered it is assumed that g b 0 of systems introduced by callier and desoer 3 the perturbation satisfies w1 w2 c now write w1gw2 g1 g1 where g1 is rational and totally unstable and g2 is stable generalizing the finite dimensional results of glover 12 this family of perturbed systems is shown to be stabilizable if and only if min g 1 the smallest hankel singular value of g 1 a finite dimensional stabilizing controller is then given by k w 2 k 1 i g 2 k 1 1 w 1 where 2 is a rational approximation of g2 such that g 2 g 2 min g 1 and k1 robustly stabilizes g1 to margin the feedback system g k will then be stable if w1 w2

the equation xr qy phi a characterization of solutions in this paper we consider the solutions of the equation xr qy phi here q r phi are given p times q m times t and p times t polynomial matrices over a field k x and y are p times m and q times t polynomial matrices which are unknown using certain recent results on the realization of matrix fraction descriptions of transfer matrices we give a characterization parametrization of all possible x y which solve this equation this also provides a system theoretic interpretation for this equation

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

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

finite dimensional controller design for infinite dimensional systems the circle criterion approach abstract the hilbert space generalization of the circle criterion is used for the finite dimensional compensator design for unstable infinite dimensional systems a design algorithm which generalizes the small gain based algorithm in 5 is presented

on the computation of induced norms for non compact hankel operators arising from distributed control problems abstract this paper considers the problem of computing the induced l 2 norm for hankel operators which are not necessarily compact in the literature such a norm has traditionally been computed by solving an infinite dimensional eigenvalue eigenfunction problem in this paper it is shown that a sequence of finite dimensional eigenvalue eigenvector problems can be solved to estimate the desired induced norm

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

a weighted mixed sensitivity h sub spl infin control design for irrational transfer matrices approximate solutions to a weighted mixed sensitivity h sub spl infin control problem for an irrational transfer matrix are obtained by solving the same problem for a reduced order rational transfer matrix upper and lower bounds for the sensitivity are obtained in terms of the sensitivity for the reduced order model and the approximation error moreover rational suboptimal controllers for the irrational transfer matrix are constructed which achieve a sensitivity close to the minimum

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

controller design for distributed systems based on hankel norm approximations a procedure for obtaining a finite dimensional reduced order model for a class of distributed systems is developed it is based on a combination of a modal approximation and an optimal hankel norm approximation and the approximation error can be accurately predicted the reduced order model can be used to design a precompensator for the original system as its performance and robust stability characteristics can be deduced from the reduced order model and finitely many modal parameters the procedure is applied to a parabolic system

robust control of flexible structures a case study a comparison is made between three partial differential equation models for a flexible beam with different types of damping and varying parameter values robust controllers can be designed to stabilize all linear systems whose transfer functions lie within a ball in the l norm given a nominal model for our flexible beam we calculate which sets of models can be stabilized by the same finite dimensional robust controller by stabilize we mean here that the poles of the system are moved to the left of the line re s 0

brief robust control of parabolic pde systems with time dependent spatial domains we synthesize robust nonlinear static output feedback controllers for systems of quasi linear parabolic partial differential equations with time dependent spatial domains and uncertain variables the controllers are successfully applied to a typical diffusion reaction process with moving boundary and uncertainty

the tracking and disturbance rejection problem for distributed parameter systems this note studies the servomechanism problem for the linear systems whose transfer function matrices belong to the algebra hat b p timesq defined and developed in 1 7 and 2 the necessary and sufficient structure of the compensator is established an internal model cascaded with a stabilizer of the loop

completion of partial matrices to contractions an n by m partially specified complex matrix is called a partial contraction if every rectangular submatrix consisting entirely of specified entries is itself a contraction necessary and sufficient condition are given for the pattern of specified entries such that any n by m partial contraction with this pattern may be completed to a full n by m contraction

the numerical range of derivations abstract let p q n be integers satisfying 1 p q n the p q numerical range of an n n complex matrix a is defined by w p q a e p uau q u unitary where for an n n complex matrix x x q denotes its q q leading principal submatrix and e p x q denotes the p th elementary symmetric function of the eigenvalues of x q when 1 p q the set reduces to the classical numerical range of a which is well known to be convex many authors have used the concept of classical numerical range to study different classes of matrices in this note we extend the results to the generalized cases besides obtaining new results we collect existing ones and give alternative proofs for some of them we also study the p q numerical radius of a defined by r p q a max w p q a

a direct approach to compensator design for distributed parameter systems we present a direct approach to finite order compensator design for distributed parameter systems i e one that is not based on reduced order modelling instead we use a parametrization around an initial compensator which displays both controller order and closed loop stability in a convenient way the main result is an existence theorem which holds for a wide class of linear time invariant systems parabolic delay damped hyperbolic the most important assumptions are bounded inputs and outputs finitely many unstable modes completeness of eigenvectors an example is included to illustrate the feasibility of our method for purposes of design

control of linear systems with fixed noncommensurate point delays the first solution is given to the fundamental open problem of stabilizability and detectability necessary sufficient conditions for internal stabilization by feedback of retarded and a large class of neutral delay differential systems with several fixed noncommensurate point delays using causal compensators observers and state feedback or dynamic output feedback which are also the same type of neutral or retarded delay differential systems with fixed point delays only our results are rank conditions on the system matrices zi f g and zif t h t evaluated at points in the complex plane and are the weakest possible generally applicable sufficient such rank conditions for stabilization of neutral systems in the light of what is known on the stability of such systems these conditions are necessary for most practical purposes the class of systems we consider includes all retarded delay differential systems with noncommensurate fixed point delays in the case of retarded systems these rank conditions are necessary and sufficient conditions for stabilization via compensators which are causal retarded delay differential systems with fixed point delays only these constitute the first full solution of these previously unsolved problems of stabilizability and detectability which together are necessary and sufficient conditions for internal stabilization by feedback for delay differential systems even in the retarded single fixed point delay case an application of our results to a problem of practical importance in control of linear systems with no delays provides a stabilization criterion interesting in itself

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

stabilization of time delay systems using finite dimensional compensators for linear time invariant systems with one or more noncommensurate time delays necessary and sufficient conditions are given for the existence of a finite dimensional stabilizing feedback compensator in particular it is shown that a stabilizable time delay system can always be stabilized using a finite dimensional compensator the problem of explicitly constructing finite dimensional stabilizing compensators is also considered

remarks on computing coprime factors for distributed parameter systems presents an indirect method for computing coprime factors for a class of linear time invariant distributed parameter systems the method overcomes the numerical instability of an existing direct decomposition method it is shown that coprime factorization can be reduced to constructing simultaneously stabilizing finite dimensional controllers for which an efficient iterative procedure is developed

numerical ranges of principal submatrices abstract it is shown that the ratio of the area of the convex hull of the fields of values of the n 1 by n 1 principal submatrices of an n by n matrix a to the area of the field of values of a is bounded below by a function of n which approaches 1 as n approaches since this convex hull is necessarily contained in the field of values of a an interpretation is that asymptotically in the dimension the field of any given matrix is filled up by the fields of the submatrices collectively some new inequalities for the eigenvalues of principal submatrices of hermitian matrices which are not implied by interlacing are employed

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

eigenvalue inequalities for products of matrix exponentials motivated by models from stochastic population biology and statistical mechanics we prove new inequalities of the form cp eae 2 cp eatr where a and b are n x n complex matrices 1 n 00 and cp is a real valued continuous function of the eigenvalues of its matrix argument for example if a is essentially nonnegative b is diagonal real and cp is the spectral radius then holds if in addition a is irreducible and b has at least two different diagonal elements then the inequality is strict the proof uses kingman s theorem on the logconvexity of the spectral radius lie s product formula and perturbation theory we conclude with conjectures

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

an approximate predictor approach to reduced order models and controllers for distributed parameter systems two reduced order digital controllers for distributed parameter systems dps are described reduced order models approximate the optimal finite past predictor and error covariance for the full system to minimize an approximation to the kullback leibler information distance klid an lqg controller based on a reduced order system model is described a reduced order controller is found to minimize the klid between the closed loop system outputs with the full and reduced order controllers noncollocated control of a flexible beam is simulated

an atlas for tridiagonal isospectral manifolds abstract let t be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix with simple spectrum we introduce bidiagonal coordinates charts defined on open dense domains forming an explicit atlas for t in contrast to the standard inverse variables consisting of eigenvalues and norming constants every matrix in t now lies in the interior of some chart domain we provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics both for continuous and discrete time

realization theoretic solution of two analytic matrix equations with application to stabilization of infinite dimensional systems a realization theoretic approach is proposed for the solutions of two matrix analytic equations arising in the control of infinite dimensional systems the bezout type equationqx ry which arises in the stabilization of infinite dimensional systems and the skew matrix equation qy xr which arises in the tracking stabilization in the presence of disturbances where q r and are given analytic matrices the realization proposed here is based on the operator theoretic machinery described by fuhrmann 1981 with this form of realization we extend to infinite dimensional systems the solutions given by emre 1980 and emre and silverman 1981 for finite dimensional systems a systematic characterization of all h2 matrix solutions of these equations is given when q r satisfies the bezout corona condition explicit formulae for the h2 solutions of the bezout type equation are given in terms of a realization and the hankel map of the given system h2 solutions are shown to be an intermediate

minimax problems on grassmann manifolds sums of eigenvalues courbures principales et energies de hartree fock grassmanniens des domaines de forme quadratique cycles de schubert le plongement de plucker fonctions de morse quadratiques caracterisation minimax de sommes de valeurs propres

quadratic forms and the graph isomorphism problem abstract we relate the graph isomorphism problem to the classical problem of equivalence of integer quadratic forms we give some necessary number theoretic conditions that the isomorphic graphs have to satisfy

special cases of a matrix exponential formula abstract some low dimension or low rank cases of a formula for a product of exponentials of matrices are studied using eigenvalue considerations counterexamples to certain natural conjectures involving a product of matrix exponentials are also given and comments relating to certain recently discovered exponential identities are included

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

an algorithmic test for checking stability of feedback spectral systems abstract given a certain class of regular spectral systems an algorithm is presented that is designed to test the stability properties of the feedback interconnection of such a system and either a bounded state feedback operator or a finite dimensional linear time invariant controller the applicability of this test is independent of the method used to design the feedback controller and is designed to provide a separate independent method for verifying whether or not the plant controller interconnection is exponentially stable the test is particularly well suited to problems involving certain partial differential equations pde with spatially varying coefficients or complicated boundary shapes

a counterexample connected with gersgorin s theorem abstract a conjectured generalization of gersgorin s eigenvalue location theorem is shown to be false

two remarks on matrix exponentials it is shown that for two square matrices a and b with algebraic elements eaeb ebea if and only if ab ba moreover the in some sense best possible perturbation inequality ea eb 2 ea 2 e a b 2 1 for a normal matrix a is proven

realization and partial fractions abstract we discuss the relation between two intrinsically different proposals that have been made in the literature concerning the representation by constant matrices of rational matrices given in fractional form it turns out that the relation is most naturally studied in the framework of partial fraction decompositions we develop the realization theory for decompositions with respect to arbitrary complementary parts of the extended complex plane which may for instance correspond to stability and instability an isomorphism is obtained which connects the spaces used in the two methods and several identities relating to the mcmillan degree are derived in a direct way finally a new computational procedure is given to obtain the partial fraction decomposition of a rational matrix given in fractional form

combinatorial matrix analysis an overview abstract we survey recent work in some components of combinatorial matrix analysis including qualitative matrix theory matrix completion problems bounding of matrix parameters the role of the longest simple circuit and attenuation of matrix products some definitional thoughts historical perspectives and unification challenges etc are offered the style is informal following the talk on which this writeup is based

borel fixed ideals and reduction number abstract the aim of this paper is to study the relationship between the reduction number and borel fixed ideals in all characteristics especially it is shown that r r i r r i lex where i lex denotes the unique lex segment ideal whose hilbert function is equal to that of i this solves a recent question by conca

poset embeddings of hilbert functions and betti numbers abstract we study inequalities between graded betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps defined earlier by us in caviglia and kummini 2013 5 we show that if graded betti numbers do not decrease when we replace ideals in an algebra by their embedded versions then the same behaviour is carried over to ring extensions as a corollary we give alternative inductive proofs of earlier results of bigatti hulett pardue mermin peeva stillman and murai we extend a hypersurface restriction theorem of herzog and popescu to the situation of embeddings we show that we can obtain the betti table of an ideal in the extension ring from the betti table of its embedded version by a sequence of consecutive cancellations we further show that the lex plus powers conjecture of evans reduces to the artinian situation