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Invariant Subspaces of Matrices with Applications

Israel Gohberg, Peter Lancaster, and Leiba Rodman
Publication Date: 
Number of Pages: 
Classics in Applied Mathematics 51
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Gizem Karaali
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Let me start by saying what this book is not: It is not a standard textbook for a second course in linear algebra. It is not a recent text (in fact it is a republication of a 1986 book). And last but not least, it is not a leisurely read; it requires a substantial amount of work from its reader, even though the language is not stuffy and the many results are motivated well and explained with an ample supply of examples.

Having said all that, I need to quickly add that this book will certainly find a valuable place in some people's libraries. There is a reason why SIAM republished it in its Classics in Applied Mathematics series. Approximately 20 years ago, Israel Gohberg, Peter Lancaster and Leiba Rodman, all very prolific and established scholars in operator theory and matrix analysis, set out to write a treatment of linear algebra in which invariant subspaces were central to the development of the whole theory of linear transformations. Most texts on invariant subspaces focus mainly on the infinite dimensional case. Based on recent developments, mostly in linear systems theory, the authors decided that a more general-purpose text assuming very few prerequisites could be written. This book, a thorough treatment of invariant subspaces, covering geometric, algebraic, analytical and topological aspects, was the outcome of that decision.

In this 2006 edition, the main text is not changed, and neither is the bibliography. The standard linear algebra books referred to have long been replaced (at least in the US classrooms) by other more recent books. In fact, all the references are articles and books written before most of today's college students were born. However, if the reader is curious about the present status of the problems dealt with in the text, the new preface, written for this new edition, will provide good leads. In this short preface, the authors discuss some recent developments which decided some of the open problems mentioned in the 1986 text and provide references to surveys which will help bridge the gaps.

Overall, the book provides a good preparation for research in some specific fields, namely, operator theory, matrix analysis, and linear systems theory. However it does not require graduate standing; an undergraduate student who has taken a semester's worth of linear algebra has the algebraic prerequisites to follow it. We should mention, though, that some exposure to complex analysis is necessary for the full appreciation of the material. Therefore undergraduate mathematics or engineering students with the right backgrounds (one course each in linear algebra and complex analysis) will find the text quite accessible.

As a snapshot of what was known in 1986 in the theory of invariant subspaces and the linear algebra that underlies it all, this is an interesting book to study. It is well-written and has plenty of exercises, so the instructor working with advanced undergraduates interested in doing research in operator theory, matrix analysis, or linear systems theory could find the book quite useful, in the classroom or for independent study. Now if I could only get beyond the weird symbol it uses for complex numbers! (Yes, I know it is a C, but…)

Gizem Karaali is assistant professor of mathematics at Pomona College.


 Preface to the Classics Edition; Preface to the First Edition; Introduction; Part One: Fundamental Properties of Invariant Subspaces and Applications. Chapter 1: Invariant Subspaces: Definitions, Examples, and First Properties; Chapter 2: Jordan Form and Invariant Subspaces; Chapter 3: Coinvariant and Semiinvariant Subspaces; Chapter 4 Jordan Form for Extensions and Completions; Chapter 5: Applications to Matrix Polynomials; Chapter 6: Invariant Subspaces for Transformations Between Different Spaces; Chapter 7: Rational Matrix Functions; Chapter 8: Linear Systems; Part Two: Algebraic Properties of Invariant Subspaces. Chapter 9: Commuting Matrices and Hyperinvariant Subspaces; Chapter 10: Description of Invariant Subspaces and Linear Transformation with the Same Invariant Subspaces; Chapter 11: Algebras of Matrices and Invariant Subspaces; Chapter 12: Real Linear Transformations; Part Three: Topological Properties of Invariant Subspaces and Stability. Chapter 13: The Metric Space of Subspaces; Chapter 14: The Metric Space of Invariant Subspaces; Chapter 15: Continuity and Stability of Invariant Subspaces; Chapter 16: Perturbations of Lattices of Invariant Subspaces with Restrictions on the Jordan Structure; Chapter 17: Applications; Part Four: Analytic Properties of Invariant Subspaces. Chapter 18: Analytic Families of Subspaces; Chapter 19: Jordan Form of Analytic Matrix Functions; Chapter 20: Applications; Appendix: List of Notations and Conventions; References; Author Index; Subject Index.