multiplicativity of lp norms for matrices abstract the l p norm and the l p operator norm of an m n complex matrix a ij are given by a p i j ij p 1 p and a p max ax p xe c n x p 1 respectively the main purpose of this paper is to investigate the multipicativity of the l p norms and their relation to the l p operator norms

lower bounds for the condition number of a real confluent vandermonde matrix lower bounds on the condition numberp vc of a real con uent vander monde matrix vc are established in terms of the dimension n or n and the largest absolute value among all nodes that deflne the con uent vander monde matrix or n and the interval that contains the nodes in particular it is proved that for any modest kmax the largest number of equal nodes p vc behaves no smaller than on 1 p 2 n and than on 1 p 2 2n if all nodes are nonnegative it is not clear whether those bounds are asymp totically sharp for modest kmax

rational stabilising commutative controllers parameterisation and characterisation of degrees of freedom the characteristic locus method constitutes a generalisation of the classical frequency response approach and as such provides a natural platform for design aimed at meeting specifications such as closed loop stability and dynamic performance however to overcome problems of sensitivity to uncertainty it is necessary to precondition the plant transfer function matrix tfm with the view to improving the orthogonality of the eigenvector functions all that remains then is to use controllers which adjust the frequency response of the eigenfunctions of the tfm while leaving the eigenvectors unaltered this implies the need for commutative controllers which may be irrational and may not be internally stabilising the present paper gives a complete characterisation of the class of stabilising rational commutative controllers and derived necessary and sufficient conditions for the existence of this class these ideas are illustrated by means of case study in which the degrees of freedom contained within the c

multiplicativity of lp norms for matrices ii abstract for 1 p let a p i 1 m j 1 n ij p 1 p be the l p norm of an m n complex a ij c m n the main purpose of this paper is to find for any p q 1 the best smallest possible constants m k n p q and m k n p q for which inequalities of the form ab p m k n p q a p b q ab p m k n p q a q b p hold for all a c m k b c k n this leads to upper bounds for inner products on c k and for ordinary l p operator norms on c m n

a normalizing precompensator for the design of effective and reliable commutative controllers sensitivity to parameter perturbation represents the main caution regarding the use of the characteristic locus method on the design of multivariable control systems the method is not effective when the condition number of the plant eigenvector matrix is high or equivalently when the plant transfer matrix differs a great deal from normality with the view to coping with this problem it is proposed in this paper a precompensator structure and in the sequel two optimization problems are formulated and solved the first one for 2 x 2 systems aims at minimizing the eigenvector matrix condition number the second one for the general m x m case is intended to make the precompensated system as normal as possible by minimizing a defined measure of normality once the precompensated system matrix has been made close to a normal one the characteristic locus method can then be applied effectively leading to reliable control systems as far as stability in face of uncertainty is concerned

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

mixed multiplicativity and lp norms for matrices abstract let cm n denote the class of m n complex matrices and let n1 n2 and n3 be arbitrary norms on cm n cm k and ck n respectively in this paper we discuss the best least positive constant minwhich satisfies n 1 ab min n 2 a n 3 b a c m k b c k n in particular for 1 p let a p i 1 m j 1 n ij p 1 p be the lp norm of a matrix a ij cm n then for arbitrary p q r such that 1 p q r we determine explicitly the best constant min for which ab p min a q b r a c m k b c k n

the use of rational eigenvector approximations in commutative controllers commutative controllers provide a convenient means for the systematic adjustment of the gain phase characteristic of generalized nyquist diagrams due to the irrational nature of the eigenvectors of transfer function matrices practical commutative controllers are based on eigenvector approximations this paper investigates the use of rational approximations highlights stability difficulties that arise in connection with unstable fixed modes and derives the necessary and sufficient conditions that overcome these difficulties the results of the paper are

on perturbations of matrix pencils with real spectra a revisit this paper continues earlier studies by bhatia and li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra a unifying framework for creating crucial perturbation equations is developed with the help of a recent result on generalized commutators involving unitary matrices new and much sharper bounds are obtained

realization algorithm for determining generalized state space representations a realization algorithm is presented that determines a generalized state space representation from a so called column pseudoproper right mfd this algorithm is a generalization of the realization algorithm given by wolovich and is proved to be controllable in the rosenbrock sense moreover the dimension of the realization is equal to the determinantal degree of the given right mfd in a generalized sense

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

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

bi c eigenvector sequences and the design of causal commutative controllers abstract the use of laurent power series expansions of the eigenvector matrix of a linear multivariable transfer function matrix g z holds the key to the physical realization of commutative controllers in general however such controllers would be anti causal it is the purpose of this paper to show that there are enough degrees of freedom in the choice of the controller eigenfunctions both to effect gain phase compensation of the frequency response of the eigenfunctions of g z and to force the resulting control law to be causal the results of the paper are shown to be superior to those possible through the use of approximate commutative controllers proposed earlier

clockwise nature of nyquist locus of stable transfer functions abstract f s is a finite proper rational function with real coefficients and all its poles in the interior of the left half plane the nyquist locus of f s is the mapping f s s j with in 0 it is shown that the nyquist locus of f s is clockwise at all in 0

h2 and h approximations for eigenvalues vector functions of transfer matrices h2 and h weighting sequence approximations to rational and irrational functions trefethen 1981 helton et al 1989 are pertinent to characteristic locus design the most effective technique combines inverse discrete fourier transforms with what amount to h2 approximations in this paper we discuss the theoretical background and develop some new h2 and h techniques the resulting algorithms can be applied to general rational irrational function and vector function approximation problems they are shown to be particularly effective in the approximation of eigenvalue vector functions of transfer function matrices the derived techniques provide some very useful tools for the characteristic locus design method

a converse to the bauer fike type theorem it is proved that the unperturbed matrix pencil is a normal pencil if there exists a small perturbation such that the equality sign holds in the bauer fike type in equality using p norms

vandermonde matrices with chebyshev nodes abstract for an n n vandermonde matrix v n j i 1 1 ij n with translated chebyshev zero nodes it is discovered that v n t admits an explicit qr decomposition with the r factor consisting of the coefficients of the translated chebyshev polynomials this decomposition then leads to an exact expression for the frobenius condition number of its submatrix v k n j i 1 1 i k 1 j n so called a rectangular vandermonde matrix bounds on individual singular value and more it is explained how these results can be used to establish asymptotically optimal lower bounds on condition numbers of real rectangular vandermonde matrices and nearly optimally conditioned real rectangular vandermonde matrices on a given interval extensions are also made for v n with nodes being zeros of any translated orthogonal polynomials other than chebyshev ones similar results hold for v n with translated chebyshev extreme nodes too owing to that v n t admits an explicit qr like decomposition close formulas of or tight bounds on the residuals are also presented for the conjugate gradient method the minimal residual method and the generalized minimal residual method on certain linear systems ax b with a having eigenvalues the same as the nodes mentioned above as a by product they yield positive definite linear systems for which the residuals by the conjugate gradient method are always comparable to the existing error bounds for all iteration steps

commutative controllers revisited parallel computation a new lease of life this paper presents a design methodology for multivariable control that combines time and frequency domain descriptions of the plant to produce appropriate compensators the proposed methodology relies on a novel convolution implementation made possible by developments in vlsi computer technology and parallel processing algorithms the result is an algorithm that completely reduces the multivariable design problem to a set of siso problems to which classical frequency response techniques can be applied the paper also establishes conditions for the existence of the necessary controller representations and proposes methods to ensure that these conditions are satisfied simulation results are presented to demonstrate the utility of the methodology

geometries of the projective matrix space in our paper on matrix mobius transformations comm algebra 9 19 1981 1913 1968 we introduced the one dimensional left projective space over the complex n n matrices p p1 mn for n 1 this space is the projective complex line p1 and so is homeomorphic to the riemann sphere we studied the topology of p and the projective mappings of p onto itself here we present some generalizations of certain parts of the euclidean the spherical and the non euclidean geometry from the scalar to the multidimensional case

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

bicausal representations and multivariable generalized predictive control abstract a change of basis from the standard to an eigenvector set has the effect of decomposing a multivariable problem into a set of characteristic scalar sub problems conformal mapping arguments can be constructed to show that satisfactory characteristic subsystem performance is a prerequisite for overall satisfactory system performance with this motivation earlier work proposed a generalization of the generalized predictive control to the multivariable case the approach was based on causal characteristic representations and this implied some limitations on the accuracy of the method in the case of plants with unstable branch points in this paper we show that under some mild conditions the characteristic subsystems admit a bicausal representation which can be made as accurate as desired bicausality does not constitute a problem in predictive control and a suitable multivariable self tuning algorithm is proposed the superiority of the derived results is demonstrated by means of a design study

norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils abstract we start by proving a lower bound for the lp operator norm of a submatrix with sufficiently large dimensions and some relations between the lp norm and the lp operator norm we then study the lp norm and the lp operator norm of certain matrices arising in estimating the eigenvalue generalized eigenvalue singular value and generalized singular value variations of matrices matrix pencils and matrix pairs finally as applications of our bounds upon the lp norm and lp operator norm of certain matrices we give several new bounds for the variations of the spectra of matrices and matrix pencils these new bounds are generalizations of the weyl lidskii theorem for hermitian matrices and the bhatia davis theorem for unitary matrices and of some results obtained by the author recently for diagonalizable pencils with real spectra

realization of singular systems via markov parameters abstract realization of singular systems via markov parameters an algorithm is developed via the markov parameters for the construction of minimal finite dimensional realizations of linear time invariant discrete time singular systems when the external description such as the impulse response matrix or the frequency response matrix is given the presented algorithm needs only simple numerical operations and it may lead to every possible realization of a given input output relation

on perturbations of matrix pencils with real spectra ii a well known result on spectral variation of a hermitian matrix due to mirsky is the following let a and a be two n x n hermitian matrices and let 1 n and 1 n be their eigenvalues arranged in ascending order then diag 1 1 n n a a for any unitarily invariant norm in this paper we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra the much studied case of definite pencils is included in this

equivalence constants for matrix norms a problem of goldberg abstract we investigate the equivalence constants for the l p coefficient norms and l q operator norms 1 p q of complex m n matrices and provide estimates which are either best possible or close to best possible

spectral variations and hadamard products some problems abstract many perturbation problems in matrix theory are related to three matrix minimization problems involving hadamard products possible solutions to the problems are conjectured the problems if solved may imply significant advances in eigenvalue generalized eigenvalue variations and perhaps in other aspects in matrix theory too