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In this chapter we review the factorization theory for the case of real matrix functions with respect to real divisors. As in the complex case the minimal factorizations are completely determined by the supporting projections of a given realization, but in this case one has the additional requirement that all linear transformations must be representable by matrices with real entries. Due to the difference between the real and complex Jordan canonical form the structure of the stable real minimal factorizations is somewhat more complicated than in the complex case. This phenomenon is also reflected by the fact that for real matrixes there is a difference between the stable and isolated invariant subspaces. [ABSTRACT FROM AUTHOR]

In this chapter we shall prove that there exist stable factorizations which are not spectral factorizations. In fact, for the finite-dimensional case we shall give a complete description of all possible stable minimal factorizations. It will also be shown that stability amounts to the same as the property of being isolated provided the underlying field is complex (which will be the case in this chapter). [ABSTRACT FROM AUTHOR]

In numerical computations of minimal factors of a given transfer function questions concerning the conditioning of the factors turn up naturally. According to the division theory developed in the previous chapters, all minimal factorizations may be obtained in an explicit way in terms of supporting projections of minimal systems. This fact allows one to reduce questions concerning the conditioning of minimal factorizations to questions concerning the stability of divisors of a system. In the present chapter we study the matter of stability of spectral divisors mainly. In this case the investigation can be carried out for finite- as well as for infinite-dimensional state spaces. The invariant subspace method employed in this chapter will also be used to prove that "spectral" solutions of an operator Riccati equation are stable. The case of minimal non-spectral factorizations will be considered in the next chapter. [ABSTRACT FROM AUTHOR]

In this chapter a connection is made between the issue of quasicomplete factorization discussed in Section 10.4 and a problem from the theory of combinatorial job scheduling. The problem in question is the so-called two machine flow shop problem (2MFSP for short) where one wants to find optimal schedules for processing jobs on two machines, given certain precedence constraints. It turns out that such problems are in correspondence (one-to-one, essentially) with the companion based rational matrix functions considered in the previous chapter. We show that the number of factors in a quasicomplete factorization of a companion based matrix function is directly related to the minimum makespan (i.e., the time needed for carrying out a optimal schedule) of the associated instance of 2MFSP. Illustrative examples are given. In one of them the (computationally fast) algorithm called Johnson's rule for 2MFSP is used to compute the quasidegree of a companion based function. [ABSTRACT FROM AUTHOR]

In this chapter results of the previous chapter are specified further for rational matrix functions of a special type, namely for the so-called companion based functions. These are characterized by the fact that they are rational matrix function having a minimal realization in which both the main matrix and the associate main matrix are (first) companions. A description of such functions is presented for the 2 × 2 case, and necessary and sufficient conditions are given for such functions to admit a complete factorization. The factorization results in this chapter are based on a detailed analysis of simultaneous reduction to complementary forms of pairs of companion matrices. [ABSTRACT FROM AUTHOR]

In this chapter we study the factorization of a proper rational m × m matrix function having the value Im at infinity into elementary factors satisfying the same constraints. These elementary factors are of McMillan degree one by definition. It turns out, by using realization, that the problem of factorizing a function in such degree one factors is intimately connected with the issue of simultaneous reduction to complementary triangular forms of pairs of matrices. We prove that factorization into elementary factors is always possible. [ABSTRACT FROM AUTHOR]

In this chapter the notion of minimal factorization of rational matrix functions, which has its origin in mathematical system theory, is introduced and analyzed. In Section 9.1 minimal factorizations are identified as those factorizations that do not admit pole zero cancellation. Canonical factorization is an example of minimal factorization but the converse is not true. In Section 9.2 we use minimal factorization to extend the notion of canonical factorization to rational matrix functions that are allowed to have poles and zeros on the curve. In Section 9.3 (the final section of this chapter) the concept of a supporting projection is extended to finite-dimensional systems that are not necessarily biproper. This allows us to prove one of the main theorems of the first section also for proper rational matrix functions of which the value at infinity is singular. [ABSTRACT FROM AUTHOR]

In this chapter finite-dimensional systems are studied in terms of the zero or pole data of their transfer functions. In the first two sections we describe the local zero and pole data, and related Jordan chains, of a meromorphic m×m matrix function of which the determinant does not vanish identically. In the third section these results are used to construct minimal realizations of rational matrix functions in terms of the zero or pole data of the function. The fourth section deals with the notions of local degree of a transfer function and local minimality of a finite-dimensional system. The global versions of these notions are studied in the final section. [ABSTRACT FROM AUTHOR]

In this chapter the notion of a minimal system is considered. If two systems are similar, then they have the same transfer function. The converse statement is not true. In fact, systems with rather different state spaces may have the same transfer function. For minimal systems this phenomenon does not occur. In Section 7.1 minimal systems are defined as systems that are controllable and observable. The latter two notions are explained in more detail for finite-dimensional systems in Section 7.2. In the finite-dimensional case the connection between a minimal system Θ and its transfer function WΘ is very close. For example in that case Θ is uniquely determined up to similarity by WΘ. This result, which is known as the state space similarity theorem, will be proved in Section 7.3. Several examples, presented in Section 7.4, show that a generalization of the finite-dimensional theory to an infinite-dimensional setting is not possible in a straightforward way. An appropriate generalization requires a further refinement of the state space theory. In Section 7.5 the notion of minimality is considered for Brodskii systems, Kreîn systems, unitary systems, monic systems, and polynomial systems. [ABSTRACT FROM AUTHOR]

As we have seen in Chapter 1 canonical factorization serves as tool to solve Wiener-Hopf integral equations and their discrete analogues, the block Toeplitz equations. In this chapter the state space factorization method developed in Chapter 2 is used to solve the problem of canonical factorization (necessary and sufficient conditions for its existence) and to derive explicit formulas for its factors. This is done in Section 6.1 for rational matrix functions. The results are applied to invert Wiener- Hopf integral equations (Section 6.2) and block Toeplitz operators (Section 6.3) with a rational matrix symbol. [ABSTRACT FROM AUTHOR]

In this chapter the state space factorization theory from Section 2.4 is presented using a different terminology. Here it will be based on the notion of an angular operator and the algebraic Riccati equation. [ABSTRACT FROM AUTHOR]

The main problem addressed in this chapter is the realization problem for operatorvalued functions. Given such a function the problem is to find a system for which the transfer function coincides with the given function. In the first section we consider rational operator functions, and in the second analytic ones. In Section 4.3 it is shown that, in a certain sense, the transfer function of a system with an invertible external operator can be reduced to a linear function, and we use this reduction to describe the singularities of the transfer function. In the final section a connection between Schur complements and linearization is described. [ABSTRACT FROM AUTHOR]

In this chapter we review the notions and results from the previous chapter for various classes of systems. Included are Brodskii systems (Section 3.1), Kreîn systems (Section 3.2), unitary systems (Section 3.3), monic systems (Section 3.4) and polynomial systems (Section 3.5). The final section (Section 3.6) concerns a change of variable in the transfer function defined by a Möbius transform. [ABSTRACT FROM AUTHOR]

In this chapter the concepts of an operator node (abstract system) and its transfer function are introduced and developed systematically. Important operations on operator nodes (abstract systems) and the corresponding operations on the associated transfer functions are studied in detail: inversion (Section 2.2), products (Section 2.3) and factorization (Section 2.4). With an eye on future applications, a detailed analysis of the relationships between the various results is given in the final section. [ABSTRACT FROM AUTHOR]

This chapter has an introductory character. It presents a number of problems involving factorization of matrix- and operator-valued functions of different types. The functions considered appear as transfer functions of input output systems (Section 1.1), as characteristic functions of Hilbert space operators (Sections 1.2 and 1.3), as monic matrix polynomials (Section 1.4) or as symbols of Wiener-Hopf and singular integral equations of various types (Sections 1.5 and 1.6). For each of these classes the corresponding factorization is described. This chapter also motivates the state space setting for solving factorization problems. [ABSTRACT FROM AUTHOR]

We revisit a formula that connects the minimal ranks of triangular parts of a matrix and its inverse and relate the result to structured rank matrices. We also address a generic minimal rank problem that was proposed by David Ingerman and Gilbert Strang. [ABSTRACT FROM AUTHOR]

For a countable number of input functions in H∞ (Dn), we find explicit analytic solutions belonging to the Orlicz-type space, $$ \exp (L^{\tfrac{1} {{2n - 1}}} ) $$. Note that $$ H^\infty (D^n ) - BMO(D^n ) \subsetneqq \exp (L^{\tfrac{1} {{2n - 1}}} ) \subsetneqq \cap _1^\infty H^p (D^n ) $$ [ABSTRACT FROM AUTHOR]

We investigate the problem of the triangular factorization of positive operators in a Hilbert space. We prove that broad classes of operators can be factorized. [ABSTRACT FROM AUTHOR]

The inverse problem for canonical differential equations is investigated for Hamiltonians with singularities. The usual notion of a spectral function is not adequate in this generality, and it is replaced by a more general notion of spectral data. The method of operator identities is used to describe a solution of the inverse problem in this setting. The solution is explicitly computable in many cases, and a number of examples are constructed. [ABSTRACT FROM AUTHOR]

It is shown that the result of Tso-Wu on the elliptical shape of the numerical range of quadratic operators holds also for the essential numerical range. The latter is described quantitatively, and based on that sufficient conditions are established under which the c-numerical range also is an ellipse. Several examples are considered, including singular integral operators with the Cauchy kernel and composition operators. [ABSTRACT FROM AUTHOR]

A fast approximate inversion algorithm is proposed for two-level Toeplitz matrices (block Toeplitz matrices with Toeplitz blocks). It applies to matrices that can be sufficiently accurately approximated by matrices of low Kronecker rank and involves a new class of tensor-displacement-rank structured (TDS) matrices. The complexity depends on the prescribed accuracy and typically is o(n) for matrices of order n. [ABSTRACT FROM AUTHOR]

We study C*-algebras Oλ which arise in dynamics of the interval exchange transformations and measured foliations on compact surfaces. Using Koebe-Morse coding of geodesic lines, we establish a bijection between Bratteli diagrams of such algebras and measured foliations. This approach allows us to apply K-theory of operator algebras to prove a strict ergodicity criterion and Keane's conjecture for the interval exchange transformations. [ABSTRACT FROM AUTHOR]

It is shown that certain transformations of multi-dimensional arrays posses unique positive solutions. These transformations are composed of linear components defined in terms of Stieltjes matrices, and semi-linear components similar to u → ku3. In particular, the analysis of the linear components extends some results of the Perron-Frobenius theory to multi-dimensional arrays. [ABSTRACT FROM AUTHOR]

We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refines previous asymptotic trace formulas by Szegő and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann. [ABSTRACT FROM AUTHOR]

We discuss several algebraic and analytic aspects of the eigenstructure (si.e., eigenvalues, eigenvectors, and generalized eigenvectors) of complex symmetric operators. In particular, we examine the relationship between the bilinear form [x,y] = induced by a conjugation C on a complex Hilbert space H and the eigenstructure of a bounded linear operator T: H → H which is C-symmetric (T = CT*C). [ABSTRACT FROM AUTHOR]

It is known that any symmetric matrix can be transformed by an explicitly computable orthogonal transformation into diagonal-plus-semiseparable form, with prescribed diagonal term. In this paper, we present perturbation bounds for such transformations, under the condition that the diagonal term is close to (part of) the spectrum of the given matrix. As an application, we provide new iterative schemes for the simultaneous refinement of the eigenvalues of a symmetric matrix, having quadratic convergence. [ABSTRACT FROM AUTHOR]

The structure of the numerical range and root zones of a class of operator functions, arising from one or two parameter polynomial operator pencils of waveguide type is studied. We construct a general model of such kind of operator pencils. In frame of this model theorems on distribution of roots and eigenvalues in some parts of root zones are proved. It is shown that, in general the numerical range and root zones are not connected but some connected parts of root zones are determined. It is proved that root zones, under some natural additional conditions which are satisfied for most of waveguide type multi-parameter spectral problems, are non-separated, i.e., they overlap. [ABSTRACT FROM AUTHOR]

It has been shown in [4, 5, 6, 31] that the Hessenberg iterates of a companion matrix under the QR iterations have low off-diagonal rank structures. Such invariant rank structures were exploited therein to design fast QR iteration algorithms for finding eigenvalues of companion matrices. These algorithms require only O(n) storage and run in O(n2) time where n is the dimensiosn of the matrix. In this paper, we propose a new O(n2) complexity QR algorithm for real companion matrices by representing the matrices in the iterations in their sequentially semi-separable (SSS) forms [9, 10]. The bulge chasing is done on the SSS form QR factors of the Hessenberg iterates. Both double shift and single shift versions are provided. Deflation and balancing are also discussed. Numerical results are presented to illustrate both high efficiency and numerical robustness of the new QR algorithm. [ABSTRACT FROM AUTHOR]

We give an extension result for positive definite operator-valued Toeplitz-Krein-Cotlar triplets defined on an interval of an ordered group. When the triplet is positive definite and measurable we give a representation result. [ABSTRACT FROM AUTHOR]

The higher order analogue of the classical Carathéodory-Julia theorem on boundary angular derivatives has been obtained in [7]. Here we study boundary interpolation problems for Schur class functions (analytic and bounded by one in the open unit disk) motivated by that result. [ABSTRACT FROM AUTHOR]

Consider the two Hilbert spaces H− and T−. Let K+: H− → T- be a bounded operator. Consider a measure ρ on H-. Denote by ρK the image of the measure ρ under K+. This paper aims to study the measure ρK assuming ρ to be the spectral measure of a Jacobi field. We present a family of operators whose spectral measure equals ρK. We state an analogue of the Wiener-Itô decomposition for ρK. Finally, we illustrate our constructions by offering a few examples and exploring a relatively transparent special case. [ABSTRACT FROM AUTHOR]

In this paper we obtain several results on the rank properties of Hadamard matrices (including Sylvester-Hadamard matrices) as well as generalized Hadamard matrices. These results are used to show that the classes of (generalized) Sylvester-Hadamard matrices and of (generalized) pseudo-noise matrices are equivalent, i.e., they can be obtained from each other by means of row/column permutations. [ABSTRACT FROM AUTHOR]

In this paper, we study stability of nonlinear dynamical systems by searching for Lyapunov functions of the form $$ \Lambda (x) = \sum\limits_{i = 1}^m {\alpha _i x_i } + \frac{1} {2}\sum\limits_{i = 1}^m {\lambda _i x_1^2 } ,\lambda _i > 0,i = 1, \ldots ,m $$, 0, i = 1,..., m, respectively xTAx, where A is a positive definite real matrix. Our search for Lyapunov functions is based on interior point algorithms for solving certain positive definite programming problems and is applicable for non-polynomial systems not considered by similar methods earlier. [ABSTRACT FROM AUTHOR]

We study inverse problems associated to first-order discrete systems in the rational case. We show in particular that every rational function strictly positive on the unit circle is the spectral function of such a system. Formulas for the coefficients of the system are given in terms of realizations of the spectral function or in terms of a realization of a spectral factor. The inverse problems associated to the scattering function and to the reflection coefficient function are also studied. An important role in the arguments is played by the state space method. We obtain formulas which are very similar to the formulas we have obtained earlier in the continuous case in our study of inverse problems associated to canonical differential expressions. [ABSTRACT FROM AUTHOR]

Testing whether a given matrix is a Toeplitz-plus-Hankel matrix amounts to the verification of a system of linear equations for the matrix entries. If the matrix dimension is large, we are forced to work with the computer and hence cannot check whether something is exactly zero. We provide bounds such that if a test quantity is smaller than the bound, then the system of linear equations may be accepted to be valid and the probability for erroneously accepting the validity of the system is smaller than a prescribed value. [ABSTRACT FROM AUTHOR]

In previous work of C.A. Tracy and the author asymptotic formulas were derived for certain operator determinants whose interest lay in the fact that quotients of them gave solutions to the cylindrical Toda equations. In the present paper we consider a more general class of operators which retain some of the properties of those cited and we find analogous asymptotics for the determinants. [ABSTRACT FROM AUTHOR]