Leon Mirsky's books

Leon Mirsky wrote two books, both of which were influential. These were An Introduction to Linear Algebra (1955) and Transversal Theory: An account of some aspects of combinatorial mathematics (1971). Below we present various information about these books including some extracts of reviews.

1. An Introduction to Linear Algebra (1955), by Leon Mirsky.
1.1. The publisher writes:

This work provides an elementary and easily readable account of linear algebra, in which the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology. The account is self-contained, and the reader is not assumed to have any previous knowledge of linear algebra. Although its accessibility makes it suitable for non-mathematicians, Professor Mirsky's book is nevertheless a systematic and rigorous development of the subject.

Part I deals with determinants, vector spaces, matrices, linear equations, and the representation of linear operators by matrices. Part II begins with the introduction of the characteristic equation and goes on to discuss unitary matrices, linear groups, functions of matrices, and diagonal and triangular canonical forms. Part II is concerned with quadratic forms and related concepts. Applications to geometry are stressed throughout; and such topics as rotation, reduction of quadrics to principal axes, and classification of quadrics are treated in some detail. An account of most of the elementary inequalities arising in the theory of matrices is also included. Among the most valuable features of the book are the numerous examples and problems at the end of each chapter, carefully selected to clarify points made in the text.

1.2. From the Preface.

My object in writing this book has been to provide an elementary and readable account of linear algebra. The book is intended mainly for students pursuing an honours course in mathematics, but I hope that the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology. The material dealt with here is not extensive and, broadly speaking, only those topics are discussed which normally form part of the honours mathematics syllabus in British universities. Within this compass I have attempted to present a systematic and rigorous development of the subject. The account is self-contained, and the reader is not assumed to have any previous knowledge of linear algebra, although some slight acquaintance with the elementary theory of determinants will be found helpful. It is not easy to estimate what level of abstractness best suits a textbook of linear algebra. Since I have aimed, above all, at simplicity of presentation I have decided on a thoroughly concrete treatment, at any rate in the initial stages of the discussion. Thus I operate throughout with real and complex numbers, and I define a vector as an ordered set of numbers and a matrix as a rectangular array of numbers.

The points of contact between linear algebra and geometry are numerous, and I have taken every opportunity of bringing them to the reader's notice. I have not, of course, sought to provide a systematic discussion of the algebraic background of geometry, but have rather concentrated on a few special topics, such as changes of the coordinate system, reduction of quadrics to principal axes, rotations in the plane and in space, and the classification of quadrics under the projective and affine groups.

The theory of matrices gives rise to many striking inequalities. The proofs of these are generally very simple, but are widely scattered throughout the literature and are often not easily accessible. I have here attempted to collect together, with proofs, all the better known inequalities of matrix theory. I have also included a brief sketch of the theory of matrix power series, a topic of considerable interest and elegance not normally dealt with in elementary textbooks. Numerous exercises are incorporated in the text. They are designed not so much to test the reader's ingenuity as to direct his attention to analogous, generalizations, alternative proofs, and so on. The reader is recommended to work through these exercises, as the results embodied in them are frequently used in the subsequent discussion. At the end of each chapter there is a series of miscellaneous problems arranged approximately in order of increasing difficulty. Some of these involve only routine calculations, others call for some manipulative skill, and yet others carry the general theory beyond the stage reached in the text. A number of these problems have been taken from recent examination papers in mathematics, and thanks for permission to use them are due to the Delegates of the Clarendon Press, the Syndics of the Cambridge University Press, and the Universities of Bristol, London, Liverpool, Manchester, and Sheffield. The number of existing books on linear algebra is large, and it is therefore difficult to make a detailed acknowledgement of sources. I ought, however, to mention Turnbull and Aitken, An Introduction to the Theory of Canonical Matrices, and MacDuffee, The Theory of Matrices, on both of which I have drawn heavily for historical references. I have received much help from a number of friends and colleagues. Professor A. G. Walker first suggested that I should write a book on linear algebra and his encouragement has been invaluable. Mr H Burkill, Mr A R Curtis, Dr C S Davis, Dr H K Farahat, Dr Christine M Hamill, Professor H A Heilbronn, Professor D G Northcott, and Professor A Oppenheim have all helped me in a variety of ways, by checking parts of the manuscript or advising me on specific points. Mr J C Shepherdson read an early version of the manuscript and his acute comments have enabled me to remove many obscurities and ambiguities; he has, in addition, given me considerable help with Chapters IX and X. The greatest debt I owe is to Dr G T Kneebone and Professor R Rado with both of whom, for several years past, I have been in the habit of discussing problems of linear algebra and their presentation to students. But for these conversations I should not have been able to write the book. Dr Kneebone has also read and criticized the manuscript at every stage of preparation and Professor Rado has supplied me with several of the proofs and problems which appear in the text. Finally, I wish to record my thanks to the officers of the Clarendon Press for their helpful co-operation.

1.3. Review by: Gaylord Maish Merriman.
Amer. Math. Monthly 63 (10) (1956), 735-736.

The straight-forward clarity of the writing is admirable. Definitions, theorems, etc., are generally stated with precision (the Corollary on p. 153 is perhaps somewhat loose in wording). The format, the composition of each page, the typography, are most attractive. ... Bridge passages like the one culling from elementary algebra material for extension to matrix algebra, and the examples of groups are enticing pedagogy; and the numerous developing exercises in the body itself of the textual material work the reader into the role of expositor. The problem lists at the end of each chapter are well, if formidably, selected. Among the many extremely well-presented portions of the book, the chapter on Matrix Analysis deserves special mention; this is tricky business, and the final application to linear differential equations is a welcome illustration.

1.4. Review by: Walter Ledermann.
The Mathematical Gazette 41 (337) (1957), 239.

Great care has been taken by the author to write an introduction to linear algebra that is elementary, readable and self-contained. Though mainly intended for Honours students in Mathematics the book also caters for readers who are more interested in physics and technology. Although the author has confined himself to topics normally included in an undergraduate course, he has planned the work on a generous scale. He has not grudged the space for some didactic asides and for a certain amount of material which belongs more to a general mathematical background than to the particular subject at hand. Also the applications, notably those to geometry, are given more than a mere mention. For these reasons the volume has become larger than one might expect of an introductory text-book, but many a reader, and especially a beginner will prefer Dr Mirsky's broad and congenial style to the concise and cold rigour which is so common in modern mathematical literature. ... There is no doubt that the subject-matter of most undergraduate courses on linear algebra is amply covered. Yet, one cannot help regretting that in a work of this size the elementary divisor theory for integral matrices is relegated to three problems at the end of a chapter and that the Jordan canonical form is not done at all. The book on the whole reads well, but it is unfortunate that the initial chapter on determinants is perhaps the weakest. The discussion of the combinatorial preliminaries is heavy and the treatment, though elementary, is rather unattractive. With his expository skill, Dr Mirsky would surely have succeeded in presenting the theory more elegantly by an alternative method, such as Weierstrass's approach which is sketched on pp. 189-92. Among the most valuable features of the book are the numerous examples and problems. The analyst will find the chapter on matrix analysis and the rich crop of inequalities particularly welcome. Altogether, this well produced volume will be appreciated by a large class of readers.

1.5. Review by Malcolm F Smiley.
Mathematical Reviews MR0074364 (17,573a)

This textbook in elementary matrix theory, although confining its discussion to the real and complex fields, introduces such basic concepts as vector space, linear operator, group, and bilinear form. The treatment does not include the Jordan canonical form nor the spectral theory for finite matrices. However, a number of elementary inequalities involving complex matrices are given and there is a lengthy discussion of power series in a matrix. Otherwise the topics covered by the text may be said to be standard. The presentation is thoroughly elementary and includes a host of exercises.

1.6. Edmund's trip to Edinburgh with Mirsky's book.

I [EFR] was an undergraduate at the University of St Andrews from 1961 to 1965. I bought Leon Mirsky's Linear Algebra book in Henderson's book shop in St Andrews, at that time the book shop that stocked university texts. One day I decided to go to Edinburgh and look for other useful books for the courses I was studying. I took the train from St Andrews (at that time the town had a railway station) and I read Mirsky's book on the 90+ minute ride to Edinburgh. It is a rather wonderful book written in a very logical manner which forms a bridge between classical matrix theory and the approach of linear transformations on vector spaces. However, I could never quite decide whether this was a happy or unhappy union of the two approaches. When I reached Edinburgh I went straight to Thin's bookshop on the North Bridge, not far from Waverly station. It was an amazing shop with all sorts of nooks and crannies where one would find treasures. I found a book which one of my applied mathematics lecturers was clearly using as basis for his lectures but he was sticking so close to the text that he had chosen not to recommend that book. I purchased that book but after paying for it I was asked if they could look in my bag. "Of course," I said, so they looked and found Mirsky's Linear Algebra book which they then accused me of stealing from their shop. It is the one and only time I have ever been accused of stealing! I managed to find the price that Henderson's had written in the book in pencil and, rather reluctantly, Thin's agreed that it wasn't their book.

2. Transversal Theory: An account of some aspects of combinatorial mathematics (1971), by Leon Mirsky.
2.1. Contents.

1. Sets, Topological Spaces, Graphs
1.1. Sets and mappings
1.2. Families
1.3. Mapping theorems and cardinal numbers
1.4. Boolean atoms
1.5. The lemmas of Zorn and Tukey
1.6. Tychonoff's theorem
1.7. Graphs

2. Hall's Theorem and the Notion of Duality
2.1. Transversals, representations, and representing sets
2.2. Proofs for the fundamental theorem for finite families
2.3. Duality

3. The Method of 'Elementary Constructions'
3.1. 'Elementary Constructions'
3.2. Transversal index
3.3. Further extensions of Hall's theorem
3.4. A self-dual variant of Hall's theorem

4. Rado's Selection Principle
4.1. Proofs of the selection principle
4.2. Transfinite form of hall's theorem
4.3. A theorem of Rado and Jung
4.4. Dilworth's decomposition theorem
4.5. Miscellaneous applications of the selection principle

5. Variants, Refinements, and Applications of Hall's Theorem
5.1. Disjoint partial transversals
5.2. Strict systems of distinct representatives
5.3. Latin rectangles
5.4. Sunsets with a prescribed pattern of overlaps

6. Independent Transversals
6.1. Pre-independence and independence
6.2. Rado's theorem on independent transversals
6.3. A characteristic property of independence structures
6.4. Finite independent partial transversals
6.5. Transversal structures and independence structures
6.6. Marginal elements
6.7. Axiomatic treatment of the rank function

7. Independence Structures and Linear Structures
7.1. A hierarchy of structures
7.2. Bases of independence spaces
7.3. Totally admissible sets
7.4. Set-theoretic models of independence structures

8. The Rank Formula of Nash-Williams
81. Sums of independence structures
8.2. Disjoint independence sets
8.3. A characterization of transversal structures
8.4. Symmetrized form of Rado's theorem on independent transversals

9. Links of Two Finite Families
9.1. The notion of link
9.2. Common representatives
9.3. The criterion of Ford and Fulkerson
9.4. Common representatives with restricted frequencies
9.5. An insertion theorem for common transversals
9.6. harder results for a single family

10. Links of Two Arbitrary Families
10.1. The theorem of Mendelsohn and Dulmage and its interpretations
10.2. Systems of representatives with repetition
10.3. Common systems of representatives with defect
10.4. Common transversals of two families
10.5. Common transversals of maximal subfamilies

11. Combinatorial Properties of Matrices
11.1. The language of matrix theory
11.2. Theorems of König, Frobenius, and Rado
11.3. Diagonals of doubly-stochastic matrices
11.4. Doubly-stochastic patterns
11.5. Existence theorems for integral matrices

12. Conclusion
12.1. Current trends in transversal theory
12.2. Future research and open questions

2.2. Review by: Richard A Brualdi.
SIAM Review 15 (2) (1973), 393-395.

The present book gives a systematic presentation of the results and set-theoretic methods of transversal theory. Practically all work on transversal theory up to 1970 is either discussed in the text or the well-done notes at the end of each chapter. No mention, aside from in the preface, is made of the methods of linear programming or network flows as they have been used in transversal theory. This is understandable, for it was the author's desire to give a systematic, self-contained account of transversal theory within a reasonable number of pages. In this he has succeeded splendidly. From the point of view of the applied and industrial mathematician, however, this probably reduces its appeal. The ideas and methods of graph theory are only occasionally referred to.

2.3. Review by E C Milner.
Mathematical reviews MR0282853 (44 #87).

Parts of the book are taken from an earlier survey article by the author and H. Perfect. The present work is more comprehensive and has been greatly influenced by research done during the last five years, but it is written in the same masterly style and with the same emphasis on the central role played by Hall's classical theorem in this branch of combinatorial mathematics. Earlier equivalents of the theorem were known, e.g., to D König, but the author considers that "it is precisely Hall's formulation that has turned out to be the master key that has unlocked many closed doors". Whether or not we agree with this emphasis, we must applaud the very elegant and lucid way in which the author develops his thesis. Particularly attractive is the manner in which the self-strengthening nature of Hall's theorem is exploited in Chapter 3. ... A major pat of the book is concerned with abstract independence (Chapters 6, 7 and 8). The terminology used here seems to be better conceived than some of the alternatives, such as "matroid" or "incidence geometry", currently being used in similar senses. ... A feature of the book is the welcome attention given to the interesting infinite versions of combinatorial problems. The reader does not need to know any sophisticated set theory to enjoy this aspect of the book since the author literally uses only one kind of transfinite argument. ... This well written book presents transversal theory as a systematic discipline and is strongly recommended to anyone interested in modern combinatorial mathematics.

JOC/EFR May 2017

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