INVARIANT SUBSPACES OF MATRICES WITH APPLICATIONS: (Second Edition)

by Israel Gohberg, Peter Lancaster, and Leiba Rodman
SIAM Classics in Applied Mathematics, 2006. 692 pages.

INDEFINITE LINEAR ALGEBRA WITH APPLICATIONS

by Israel Gohberg, Peter Lancaster, and Leiba Rodman
Birkhauser, Basel, 2005. 357 pages.
ISBN 3-7643-7349-0

This book is written for graduate students, engineers, scientists and mathematicians. It starts with the theory of subspaces and orthogonalization and then goes on to the theory of matrices, perturbation and stability theory. All of this material is developed in the context of linear spaces with an indefinite inner product. The book also includes applications of the theory to the study of matrix polynomials with selfadjoint coefficients, to differential and difference equations (of first and higher order) with constant coefficients, and to algebraic Riccati equations.

List of Chapters

  1. Introduction and outline
  2. Indefinite Inner Products
  3. Orthogonalization and Orthogonal Polynomials
  4. Classes of Linear Transformations
  5. Canonical Forms
  6. Real H-Selfadjoint Matrices
  7. Functions of H-Selfadjoint Matrices
  8. H-Normal Matrices
  9. General Perturbations. Stability of Diagonalizable Matrices
  10. Definite Invariant Subspaces
  11. Differential Equations of First Order
  12. Matrix Polynomials
  13. Differential and Difference Equations of Higher Order
  14. Algebraic Riccati Equations
  15. Appendix: Topics from Linear Algebra

LAMBDA-MATRICES AND VIBRATING SYSTEMS: (Second Edition)

by Peter Lancaster
Dover, Mineola, 2002. 193 pages.>br> ISBN 0-486-42546-0

(From the preface to the first edition:)

The author's primary purpose in writing this book is to present under one cover several aspects and solutions of the problems of linear vibrating systems with a finite number of degrees of freedom, together with a careful account of that part of the theory of matrices required to deal with these problems efficiently. The treatment is intended to be mathematically sound and yet involve the reader in a minimum of mathematical abstraction. The results of the later chapters are then more readily available to those engaged in the practical analysis of vibrating systems. Indeed, the reader may prefer to dip straight into the later chapters on applications and refer back to the first four chapters for enlargement on theoretical issues as this becomes necessary.

List of Chapters

  1. A Sketch of some Matrix Theory
  2. Regular Pencils of Matrices and Eigenvalue Problems
  3. Lambda-Matrices, I
  4. Lambda-Matrices, II
  5. Some Numerical Methods for Lambda-Matrices
  6. Ordinary Differential Equations with Constant Coefficients
  7. The Theory of Vibrating Systems
  8. On the Theory of Resonance Testing
  9. Further Results for Systemns with Damping

TRANSFORM METHODS IN APPLIED MATHEMATICS: AN INTRODUCTION

by Peter Lancaster and Kestutis Salkauskas
John Wiley, New York, 1996. 332 pages.
ISBN 0-471-00810-9

This book contains an introductory course of study in the theory and practice of continuous and discrete transforms. It developed from a course given by the authors over several years which was required for geophysics majors. The most unusual feature is the presentation of what is generally seen as advanced material at a third-year undergraduate level. In order to provide users with an early introduction to important techniques, and to give math majors an early introduction to the same topics, heuristic argument and discussion are combined with careful mathematical statements.
In order to make the book accessible to a variety of audiences there are five appendices giving careful reviews of necessary topics from algebra and calculus.

List of Chapters

  1. The Laplace Transform
  2. Elementary Functions of a Complex Variable
  3. Fourier Series and the Discrete Fourier Transform
  4. Complex Integrals and Power Series
  5. The z-Transform and Discrete Filters
  6. The Fourier Transform and Continuous Filters
  7. Wavelets

List of Appendices

  1. Vectors, Matrices, and Linear Spaces
  2. Classes of Functions
  3. Two Important Limits
  4. Improper Integrals
  5. Complex Numbers

ALGEBRAIC RICCATI EQUATIONS

by Peter Lancaster and Leiba Rodman
Oxford University Press, 1995. 480 pages.
ISBN 0-19-853795-6

In this monograph we make a self-contained survey of the available knowledge (at the time of writing) of the nature of the solution sets of algebraic Riccati equations (ARE) of the two principal types (CARE and DARE). The chapter headings are:

PART I - MATRIX THEORY

Chapter 1: Preliminaries from the theory of matrices
Chapter 2: Indefinite scalar products
Chapter 3: Skew-symmetric scalar products
Chapter 4: Matrix theory and control
Chapter 5: Linear matrix equations
Chapter 6: Rational matrix functions

PART II - CONTINUOUS ALGEBRAIC RICCATI EQUATIONS

Chapter 7: Geometric theory: the complex case
Chapter 8: Geometric theory: the real case
Chapter 9: Constructive existence and comparison theorems
Chapter 10: Hermitian solutions and factorizations of rational matrix functions
Chapter 11: Perturbation theory

PART III - DISCRETE ALGEBRAIC RICCATI EQUATIONS

Chapter 12: Geometric theory for the discrete algebraic Riccati equation
Chapter 13: Constructive existence and comparison theorems
Chapter 14: Perturbation theory for the discrete algebraic Riccati equations
Chapter 15: Discrete algebraic Riccati equations and matrix pencils

PART IV - APPLICATIONS AND CONNECTIONS

Chapter 16: Linear-quadratic regulator problems
Chapter 17: The discrete Kalman filter
Chapter 18: The total least-squares technique
Chapter 19: Canonical factorization
Chapter 20: H-infinity control problems
Chapter 21: Contractive rational matrix functions
Chapter 22: The matrix sign-function
Chapter 23: The structured stability radius

Lectures on LINEAR ALGEBRA, CONTROL, AND STABILITY

by Peter Lancaster with Dan Veiner
Department of mathematics and Statistics, University of Calgary, 1999. 72 pages.

Chapter 1: Basic Materials
Chapter 2: Stability and Canonical Forms
Chapter 3: Realization of Rational Matrix Functions
Chapter 4: Balanced Realization and Model Reduction
Chapter 5: Algebraic Riccati Equations for Continuous Systems
Chapter 6: Algebraic Riccati Equations for Continuous Systems
Chapter 7: Stability
REFERENCES