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Abstract: Systems and control theory has long been a rich source of problems for the numerical linear algebra community. In many problems, conditions on analytic functions of a complex variable are usually evaluated by solving a special generalized eigenvalue problem. In this paper we develop a general framework for studying such problems. We show that for these problems, solutions can be obtained by either solving a generalized eigenvalue problem, or by solving an equivalent eigenvalue problem. A consequence of this observation is that these problems can always be solved by finding the eigenvalues of a Hamiltonian (or discrete-time counterpart) matrix, even in cases where an associated Hamiltonian matrix, cannot (normally) be defined. We also derive a number of new compact tests for determining whether or not a transfer function matrix is strictly positive real. These tests, which are of independent interest due to the fact that many problems can be recast as SPR problems, are defined even in the case when the matrix is singular, and can be formulated without requiring inversion of the system matrix . [Copyright &y& Elsevier]

In this article, we give necessary and sufficient spectral conditions for various notions of strict positive realness for single-input single-output descriptor systems. These conditions only require calculation of eigenvalues of a single matrix. A characterisation of a Kalman-Yacubovich-Popov-like lemma for descriptor systems is also derived, and its implications for the stability of a class of switched descriptor systems are briefly discussed. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of obtaining time-domain multiplier stability criteria. We show that in the case of single-input single-output systems, time-domain versions of classical frequency domain multiplier criteria can be obtained in a straightforward manner. In many cases, the time-domain criteria are more compact and more readily verified than the original criteria. Our results show that conventional constraints on the frequency response of a transfer function become simple eigenvalue evaluations in the time-domain. [ABSTRACT FROM AUTHOR]

In this paper we present sufficient conditions for the stability of a number of classes of time-varying systems. These results are of interest for two reasons. Firstly, the conditions presented take the form of matrix pencil eigenvalue criteria, and are therefore co-ordinate independent and easily verifiable. Secondly, we show a direct relationship between the form of our criteria, and the choice of Lyapunov function used to demonstrate stability (quadratic or unic). These results indicate the importance of the region of the complex plane known as the 45°-Region. [ABSTRACT FROM AUTHOR]

Finite, discrete, time-homogeneous Markov chains are frequently used as a simple mathematical model of real-world dynamical systems. In many such applications, an analysis of clustering behaviour in the states of the system is desirable, and it is well-known that the eigendecomposition of the transition matrix A of the Markov chain can provide such insight. Clustering methods based on the sign pattern in the second eigenvector of A are frequently used when A has dominant eigenvalues that are real. In this paper, we present a method to include an analysis for complex eigenvalues of A which are close to 1. Since a real spectrum is not guaranteed in most applications, this is a valuable result in the area of spectral clustering in Markov chains. [ABSTRACT FROM AUTHOR]

In this paper we give necessary and sufficient conditions for weak and strong quadratic stability of a class of switched linear systems consisting of two subsystems, associated with symmetric transfer function matrices. These conditions can simply be tested by checking the eigenvalues of the product of two subsystem matrices. This result is an extension of the result by Shorten and Narendra for strong quadratic stability, and the result by Shorten et al. on weak quadratic stability for switched linear systems. Examples are given to illustrate the usefulness of our results. [ABSTRACT FROM AUTHOR]

Abstract: We consider a convex combination of matrices that arise in the study of communication networks and the corresponding convex combination of Kronecker squares of these matrices. We show that the spectrum of the first convex combination is contained in the spectrum of the second set and that the second largest eigenvalues coincide. [Copyright &y& Elsevier]