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A Concise Text on Advanced Linear Algebra

Yisong Yang
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

The last few years have seen the publication of several textbooks with the phrase “advanced linear algebra” as all or part of the title: examples include Loehr’s Advanced Linear Algebra, Cooperstein’s Advanced Linear Algebra, Roman’s Advanced Linear Algebra and Weintraub’s A Guide to Advanced Linear Algebra. There are also books, like Lax’s Linear Algebra and its Applications, that do not use the term “advanced”, but are nonetheless. Although each of these books has its own personality (Loehr’s text is the largest and most comprehensive, and is heavily algebraic; Weintraub’s is the most concise and narrowly focused: Roman’s has some fairly nonstandard material on, for example, the umbral calculus and also affine geometry from a linear algebraic point of view), they also seem to have some features in common. Each begins with a review of basic (no pun intended) linear algebra (vector spaces, linear transformations, etc.), and each treats certain topics not generally taught in an introductory course as apparently being essential to a second, advanced, course in the subject: there seems to be agreement, for example, that quadratic forms and the Jordan form are critical topics for a book with this title.

The book under review comports with this general understanding but also adds some interesting features of its own that distinguish it from others with a similar title. The first four chapters — titled, respectively, vector spaces, linear mappings, determinants, and scalar products — cover topics that are generally at least mentioned in a standard introductory course, but the treatment here is at a higher level than is the usual introductory account. For example, vector spaces are defined here over arbitrary fields, not just the real and complex numbers, and some topics that may escape notice in an introductory course (e.g., the dual space, norm of a linear transformation, matrix exponential) are introduced early here. Unfortunately, the term “basis” is defined only for finite-dimensional spaces; I can understand why the author might not want to get involved with the set-theoretic chicanery that is required to prove the existence of a basis for any vector space, but surely in an advanced text on liner algebra, these ideas should at least be mentioned.

The treatment of determinants in chapter 3 is also somewhat unusual. Determinants are defined in a standard way in terms of cofactor expansions, but this definition is prefaced by a fairly extensive motivational section on the index of a vector field (developed in sufficient detail to allow a proof of the Fundamental Theorem of Algebra).

The motivation for determinants sets the tone for a recurring theme in the book, namely the use of analysis to motivate, and sometimes prove, results. Later in this chapter, for example, the author proves the Cayley-Hamilton theorem — that any square matrix is a root of its characteristic polynomial — in several different ways; one way consists of proving it first for matrices with distinct eigenvalues and then arguing that such matrices are dense in the set of all matrices. Later in the book, the compactness of the unit sphere in n-dimensional space is used to prove the orthogonal diagonalizability of real symmetric matrices.

The remaining chapters of the book address topics beyond those generally discussed in a standard first course. Chapters 5 and 6 discuss bilinear forms and quadratic forms over, respectively, the fields of real and complex numbers. Topics covered here include normal and self-adjoint mappings, and the Cholesky decomposition of both real and complex matrices.

Chapter 7 begins with a quick look at the set of polynomials over a field. Assuming the division algorithm for polynomials, the author defines an ideal in this set (he avoids the word “ring”) and proves that any such ideal is principal. This is then used to discuss diagonalizability, the minimal polynomial, and the Jordan canonical form. (The rational canonical form is not discussed.)

Chapter 8 is titled “Special Topics” and consists of four independent sections, covering, respectively: the Schur decomposition (given any operator on a vector space with positive definite scalar product, there is an orthonormal basis with respect to which the operator is upper triangular); the classification of skew-symmetric bilinear forms; the Perron-Frobenius theorem for nonnegative matrices; and stochastic matrices.

The last (and most unusual) chapter of the book is an introduction to quantum mechanics from the standpoint of linear algebra. The author keeps things simple by making everything finite-dimensional: the “state space” is the vector space of n-tuples of complex numbers rather than a Hilbert space and the “observables” are Hermitian n × n matrices rather than unbounded operators. Even with these simplifications, I suspect that the people who will benefit most from this chapter are people with some prior understanding of physics; nevertheless, the inclusion of a chapter like this is an interesting idea and could provide some interesting lectures in class.

Each chapter of the book is divided into sections, and each section ends with a collection of exercises. Many of these call for proofs, and some of them (less than one quarter, I would estimate) have solutions appearing in an appendix of roughly 45 pages.

One nice feature of the book is that the author makes a real effort to accommodate people with potentially differing viewpoints. Just as people who teach abstract algebra are divided into the rings-first and groups-first camps, there are differing opinions as to whether linear algebra should be matrix or transformation oriented; proponents of either view should find things of interest in this text. Likewise, those who think an advanced course should be theoretical will find much here to like, but (see particularly the chapter on quantum mechanics) those who like nontrivial applications are not ignored.

The book claims that it is intended for senior undergraduate or beginning graduate students, but I would hesitate to recommend it to any but the very strongest of undergraduates. The author’s writing style is fairly concise and terse; too much so, I think, for an average undergraduate audience. For example, the chapter on the Jordan form covers all the material described above, and more, in less than 20 pages of text. There is also, throughout the text, a dearth of examples; the author never, for example, actually computes the Jordan form of any specific matrix.

As a text for a graduate course, this book may fare better. The topic coverage is fairly similar to that in Hoffman and Kunze’s classic book (herafter denoted HK), although HK does not discuss quantum mechanics but does discuss groups of matrices preserving bilinear forms. One advantage this book may have over HK is that the latter book (which, back in the day, was actually intended for an undergraduate audience), spends a lot of time on elementary material that an instructor of a graduate course may wish to cover rapidly; Yang’s text does allow a quicker path to the more advanced stuff. Notwithstanding this, I retain a certain sentimental fondness for HK; like Michael Berg, who reviewed HK in this column, I read that book as a graduate student studying for qualifying exams, and was then, and still am, exceptionally impressed with both the exposition and the exercises.

There are some other topics that a graduate instructor might wish to cover that are not treated in the book under review. Specifically, there is no discussion of multilinear algebra and tensors, and nothing much on the numerical side of linear algebra. Even without turning this into a numeral linear algebra text, there are some basic topics (the Rayleigh-Ritz theorem, for example) that some people might wish to see included in a graduate-level book.

Some of the mathematical terminology in Yang’s text seemed at times to be nonstandard; for example, the author doesn’t use the phrase “inner product” but instead uses “scalar product”; however, his “scalar product” is simply a symmetric bilinear function. For what I would call inner products, he uses the term “positive definite scalar product”. The terms “self-adjoint” and “Hermitian” are not synonyms; the latter word is defined for matrices, the former for mappings. Likewise, on at least one occasion the author spoke of two transformations as being commutative rather than commuting. I don’t see this as any real problem, however.

So, to summarize: while I’m not sure I would recommend this book over Hoffman and Kunze, or Lax, this is certainly one that merits perusal by an instructor of a graduate course in linear algebra.

Mark Hunacek ( teaches mathematics at Iowa State University. 

Notation and convention
1. Vector spaces
2. Linear mappings
3. Determinants
4. Scalar products
5. Real quadratic forms and self-adjoint mappings
6. Complex quadratic forms and self-adjoint mappings
7. Jordan decomposition
8. Selected topics
9. Excursion: quantum mechanics in a nutshell
Solutions to selected problems
Bibliographic notes