An Overview of Matrix Factorization Theory and Operator Applications

State space methods for solving tangential interpolation problems for matrix functions in a given class have been developed extensively. These are interpolation problems in which the values in certain directions of the interpolant matrix functions, and several of their derivatives, are prescribed at given points in the complex plane. For the most part, the matrix functions under consideration are rational functions or polynomials (perhaps with symmetries); however, other important classes of matrix functions have been studied as well, notably, the Nevanlinna classes on the unit disk and the upper half-plane, and the Hardy class H/sub 2/ on the unit disk. This development is motivated by many important applications in systems and control. We present some recent results and open problems in this area.

Explicit formulas for the (generalized) inverse and criteria of invertibility are given for block Toeplitz and Wiener-Hopf type operators. We consider operators with symbols defined on a curve composed of several non-intersecting simple closed contours. Also criteria and explicit formulas for canonical factorization of matrix functions relative to a compound contour are presented. The matrix functions we work with are rational on each of the compounding contours but the rational expressions may vary from contour to contour. We use realizations for each of the rational expressions and the final results are stated in terms of invertibility properties of a certain finite matrix called indicator, which is built from the realizations. The analysis does not depend on finite dimensionality and is carried out for operator valued symbols.

Diagonalizability of operators occurring in linear transport theory is discussed from a general point of view. In particular, the operator A−1T, where T is the multiplication operator in L2 (−1, 1) and A is given by a formula of the type A f = f − Σ o n aj pj , is investigated. Diagonalization of this operator which is connected with one-group neutron transport is carried out in the general case that the coefficients aj are arbitrary complex numbers. Also, a peculiarity of multi-group theory, where the operator involved has a multiple continuous spectrum, is pointed out. A correct interpretation of the main result of that theory is provided.

The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix M depending on the Weyl coordinates ρ and v, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the (ρ, v) plane on which some elements of M(ρ, v) tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.

Matrix Riemann–Hilbert problems and factorization on Riemann surfaces

Toeplitz and Laurent operators play a basic role in systems theory. For example, a lower triangular Toeplitz matrix can be viewed as an input-output map for a linear causal time invariant system. First we introduce the Fourier transform. Then we will study Toeplitz and Laurent operators. The Fourier transform will be used to turn Laurent operators into multiplication operators and visa versa.

‎For a class of continuous functions including complex polynomials in $z$ and $\bar{z},$ we show that‎ ‎the corresponding Toeplitz operator on the Bergman space of the unit disk‎ ‎can be expressed as a quotient of certain differential operators with holomorphic coefficients‎. ‎This enables us to obtain several nontrivial operator theoretic results about such Toeplitz operators‎, ‎including a new criterion for invertibility of a Toeplitz operator for a class of harmonic symbols‎. ‎

A critical comparison is made of different methods to obtain the mobility in linear electron transport theory. An approximation to the Kubo formula is studied, a calculation based on the displaced Maxwellian distribution function, and a method introduced in path integral calculations. It is shown that the three methods are equivalent: they lead to the same resistance in the weak scattering limit, for small electric fields. Special attention is given to the weak coupling Frohlich polaron mobility.

A novel method for factorizing the transfer function matrix of a multivariable system described by the triple (A, B, C) is presented. The factorization consists of a product of two rational function matrices, one of which is the inverse of a rational function matrix. Various properties of the factorization are given and the factorization procedure is illustrated by an example.

Let M be a complete Riemannian manifold and assume that M is partitioned by a hypersurface N . In this paper we introduce a novel class of functions C_{\mathrm{w}}(M) on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of \phi that belongs to C_{\mathrm{w}}(M) we construct an index class Ind (\phi , D) in K_{1} -group of the Roe algebra of M by using the Kasparov product. It is supposed to be a counterpart of Roe's odd index class. We finally prove that Connes' pairing of \mathrm{Ind}(\phi , D) and Roe's cyclic 1 -cocycle is equal to the Fredholm index of a Toeplitz operator on N . This is an extension of the Roe–Higson index theorem to even-dimensional partitioned manifolds.

A number of problems in the control of linear feedback systems can be reduced to the following: we are given three stable rational matrix functions K, ϕ, ψ of sizes p 1 x q 1, p 1 x q 2 and p 2 x q 1 respectively, and seek a stable rational q 2 x p 2 matrix function S so as to minimize ¦K + ϕSψ¦∞. We assume that p 1 ≥ q 2, p 2≤q 1 and that ϕ and ψ have maximal rank (q 2 and p 2 respectively) on the jω-axis. Given a tolerance level μ sufficiently large, we obtain a linear fractional map G → F = [0 11 G + 0 12, 0 13][0 21 G + 0 22, 0 23]−1 such that F = K + ϕSψ with S stable and ¦F ∞ ≤ μ if and only if G is a stable q 2 x p 2 matrix function with ¦G¦∞ ≤ 1. The computation of 0 = [0 ij ] (1 ≤ i ≤ 2, 1 ≤ j ≤ 3) reduces to solving a pair of symmetric Wiener–Hopf factorization problems. For the special case where ϕ = [I q2, 0]T, ψ = [I p2, 0] (and K not necessarily stable) to which the general case can be reduced, we provide explicit state-space formulae for 0 in terms of a state-space realization of K and the ...

Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators

In this chapter we treat the problem of factorizing a non-proper rational matrix function. The realization used in the earlier chapters is replaced by $$ V(\lambda ) = I + C(\lambda G - A)^{ - 1} B. $$ Here I=I m is the m × m identity matrix, A and G are square matrices of order n say, and the matrices C and B are of sizes m × n and n × m, respectively. Any rational m × m matrix function W, proper or non-proper, admits such a representation. The representation (4.1) allows us to extend the results obtained in the previous chapter to arbitrary rational matrix functions. As an application we treat the problem to invert a singular integral operator with a rational matrix symbol.

This paper describes a general visualization model for 3D scalar data fields based on linear transport theory which contains common volume rendering models as special cases and/or approximations. The concept of "virtual" particles for the extraction of information from data fields is introduced. The role of different types of interaction of the data field with those particles such as absorption, scattering, source and color shift are discussed and demonstrated.Special attention is given to possible tools for the enhancement of interesting data features. Random distortions or noise of the data field are mapped onto the image plane in a well-defined way such that picture processing methods can be used to reconstruct the appearance of the undistorted data set. Random texturing can provide visual insights as to the magnitude and distribution of deviations of related data fields, e.g., originating from analytic models and measurements, or in the noise content of a given data field. Hidden symmetries of a data set can often be identified visually by allowing it to interact with a preselected beam of "physical" particles with the attendant appearance of characteristic structural effects such as channeling.

We consider boundary-layer structures that arise in connection with the transport of neutral particles (e.g., photons or neutrons) through a participating medium. Such boundary-layer structures were previously identified by the authors in certain particular cases [Phys. Rev. E 104, L032801 (2021)2470-004510.1103/PhysRevE.104.L032801]. Extending the previous work to anisotropic scattering and general Fresnel boundary conditions, this contribution presents computational algorithms which (1) resolve the aforementioned layers as well as previously unreported boundary layers associated with Fresnel boundary transmission and reflection, and (2) yield accurate simulations at fixed computational cost for transport under phase functions with arbitrarily strong anisotropy. The present paper additionally includes (3) Mathematical proofs which justify the numerical methods proposed for resolution of boundary-layer structures. The impact of the new theory on algorithmic performance is demonstrated through a series of 1D computational benchmarks that emulate typical photon- and neutron-transport applications such as, e.g., optical tomography, and nuclear reactor analysis and design. Experimental results for transmission of photons through turbid media are presented, exhibiting close agreement between simulated and experimental data. As illustrated by means of a variety of numerical results, the proposed boundary-layer-based approach tackles transport problems with unprecedented accuracy and efficiency.