Reconstruction of a rational nonsquare matrix function from local data

In many problems the local zero-pole structure (i.e. locations of zeros and poles together with their orders) of a scalar rational functionw is a key piece of structure. Knowledge of the order of the pole or zero of the rational functionw at the point is equivalent to knowledge of the -module (where is the space of rational functions analytic at ). For the more intricate case of a rationalp×m matrix functionW, we consider the structure of the module as the appropriate analogue of zero-pole structure (location of zeros and poles together with directional information), where is the set of column vectors of heightm with entries equal to rational functions which are analytic at . Modules of the form in turn can be explicitly parametrized in terms of a collection of matrices (C ,A ,B ,B , ) together with a certain row-reduced(p–m)×m matrix polynomialP(z) (which is independent of ) which satisfy certain normalization and consistency conditions. We therefore define the collection (C ,A ,Z ,B , ,P(z)) to be the local spectral data set of the rational matrix functionW at . We discuss the direct problem of how to compute the local spectral data explicitly from a realizationW(z)=D+C(z–A) ^–1 B forW and solve the inverse problem of classifying which collections (C ,A ,Z ,B , ,P(z)) satisfying the local consistency and normalization conditions arise as the local spectral data sets of some rational matrix functionW. Earlier work in the literature handles the case whereW is square with nonzero determinant.     

Local minimal factorizations of rational matrix functions in terms of null and pole data: Formulas for factors

It is known that local minimal factorizations of a rational matrix function can be described in terms of local null and pole data (expressed in the form of left null-pole triples and their corestrictions) of this function. In this paper we give formulas for the factors in a local minimal factorization that corresponds to a given corestriction of the left null-pole triple.The first version of this paper was written while the second author visited the College of William and Mary.Partially supported by the NSF grant DMS-8802836 and by the Binational United States-Israel Foundation grant.     

Factorizations of and extensions to J-unitary rational matrix functions on the unit circle

This paper concerns two topics: (1) minimal factorizations in the class ofJ-unitary rational matrix functions on the unit circle and (2) completions of contractive rational matrix functions on the unit circle to two by two block unitary rational matrix functions which do not increase the McMillan degree. The results are given in terms of a special realization which does not require any additional properties at zero and at infinity. The unitary completion result may be viewed as a generalization of Darlington synthesis.     

Invariance properties of a triple matrix product involving generalized inverses

Given complex-valued matrices A, B and C of appropriate dimensions, this paper investigates certain invariance properties of the product AXC with respect to the choice of X, where X is a generalized inverse of B. Different types of generalized inverses are taken into account. The purpose of the paper is three-fold: First, to review known results scattered in the literature, second, to demonstrate the connection between invariance properties and the concept of extremal ranks of matrices, and third, to add new results related to the topic.     

Jordan structures of alternating matrix polynomials

Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials.     

Localic real functions: A general setting

In pointfree topology the lattice-ordered ring of all continuous real functions on a frame L has not been a part of the lattice of all lower (or upper) semicontinuous real functions on L just because all those continuities involve different domains. This paper demonstrates a framework in which all those continuous and semicontinuous functions arise (up to isomorphism) as members of the lattice-ordered ring of all frame homomorphisms from the frame L(R) of reals into S(L), the dual of the co-frame of all sublocales of L. The lattice-ordered ring is a pointfree counterpart of the ring R^X with X a topological space. We thus have a pointfree analogue of the concept of an arbitrarynot necessarily (semi) continuous real function on L. One feature of this remarkable conception is that one eventually has: lower semicontinuous + upper semicontinuous = continuous. We document its importance by showing how nicely can the insertion, extension and regularization theorems, proved earlier by these authors, be recast in the new setting.     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

Compatible relations on filters and stability of local topological properties under supremum and product

An abstract scheme using particular types of relations on filters leads to general unifying results on stability under supremum and product of local topological properties. We present applications for Fréchetness, strong Fréchetness, countable tightness and countable fan-tightness, some of which recover or refine classical results, some of which are new. The reader may find other applications as well.     

On some generalizations of the Vandermonde matrix and their relations with the Euler beta-function

A multiple Vandermonde matrix which, besides the powers of variable, also contains their derivatives is introduced and an explicit expression of its determinant is obtained. for the case of arbitrary real powers, when the variables are positive, it is proved that such generalized multiple Vandermonde matrix is positive definite for appropriate enumerations of rows and columns. As an application of these results, some relations are obtained which in the one-dimensional case give the well-known formula for the Euler betafunction.     

Algebrability of some families of Darboux-like functions

We prove that the class of all almost continuous in the sense of Stallings functions possessing the Perfect Road Property and the class of all almost continuous functions without the Perfect Road Property contain a 2^c$2^{\mathfrak{c}}$-generated free algebras. We prove also that the family of all extendable functions contains a 2^c$2^{\mathfrak{c}}$-dimensional linear space.     

Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band

Let S be a band in Z² bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ¹ sequence of matrices {c_j}_jS we prove that there is a positive definite matrix function f in the Wiener algebra on the bitorus such that the Fourier coefficients equal c_k for k S. A parameterization is obtained for the set of all positive extensions f of {c_j}_jS. We also prove that among all matrix functions with these properties, there exists a distinguished one that maximizes the entropy. A formula is given for this distinguished matrix function. The results are interpreted in the context of spectral estimation of ARMA processes.     

Perturbation of analytic hermitian matrix functions

The behaviour of real eigenvalues of selfadjoint analytic matrix valued functions under small selfadjoint analytic perturbations is studied. Attention is paid mainly to the case when the perturbation is definite (or semidefi-nite). Earlier results of the authors concerning matrix polynomials of first degree are extended to the case of analytic functions.     

Quasicomplete factorizations of rational matrix functions

It is shown that within the class ofn×n rational matrix functions which are analytic at infinity with valueW()=I _n, any rational matrix functionW is the productW=W ₁...W _p of rational matrix functionsW ₁,...,W _p of McMillan degree one. Furthermore, such a factorization can be established with a number of factors not exceeding 2(W)–1, where (W) denotes the McMillan degree ofW.     

Asymptotics of unitary and orthogonal matrix integrals

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large N limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the orthogonal group.     

Separating sets by lower and upper semicontinuous functions with a closed graph

In this paper we characterize the pairs (A^?, A⁺) of disjoint subsets of perfectly normal topological space which can be separated by a lower and an upper semicontinuous function with a closed graph.     

Determinant of a matrix that commutes with a Jordan matrix

Given a Jordan matrix J, we obtain an explicit formula for the determinant of any matrix T that commutes with it.