The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

This paper is a brief historical review of linear singular systems, followed by a survey of results on their solution and properties. The frequency domain and time domain approaches are discussed together to sketch an overall picture of the current status of the theory.

A survey of linear singular systems

A tutorial on the geometric analysis of linear time-invariant implicit systems

The semistate theory of circuits and systems is reviewed with regard to its applications. Among these latter discussed are electronic circuit design and analysis, hysteresis, knot generation, neural networks, and constrained robot characterizations.

Some circuits and systems applications of semistate theory

A standard way of dealing with a matrix polynomial $P(\lambda)$ is to convert it into an equivalent matrix pencil—a process known as linearization. For any regular matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when $P(\lambda)$ is a singular square matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial $P(\lambda)$ from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications.

/pdf/fiedler-companion-linearizations-and-the-recovery-of-minimal-zsp6fu105l.pdf

Fiedler Companion Linearizations and the Recovery of Minimal Indices

A new characterization of infinite poles and zeros in terms of a natural extension of the idea of infinite elementary divisors of a matrix pencil to matrix polynomials is developed. It is used to s...

Infinite elementary divisors of a matrix polynomial and implications

Generalised Autonomous Dynamical Systems, Algebraic Duality and Geometric Theory

A new transformation of matrix pencils is proposed and its theory formally developed. Its restriction to generalized state-space system-matrix pencils is shown to be an appropriate generalization of the usual state-space transformation and to represent a closed-form expression in system-matrix terms of Verghese's concept of strong equivalence of such pencils.

Transformations of matrix pencils and implications in linear systems theory

A two-level algorithm which reduces a general 2-D polynomial matrix to an equivalent matrix pencil form is presented. The exact nature of the equivalence transformation connecting the matrix with its associated pencil is established.

Matrix pencil equivalents of a general 2-D polynomial matrix

A fundamental form of equivalence of polynomial matrix descriptions of linear multivariable systems is defined, based on the existence of a bijective map between the finite and infinite solution sets of the differential equations describing the two systems. The connection with the system matrix relationship of full system equivalence is established. >

G.E. Hayton

Papers

Infinite elementary divisors of a matrix polynomial and implications

Generalised Autonomous Dynamical Systems, Algebraic Duality and Geometric Theory

Transformations of matrix pencils and implications in linear systems theory

Matrix pencil equivalents of a general 2-D polynomial matrix

A fundamental notion of equivalence for linear multivariable systems