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Applied Linear Algebra: The Decoupling Principle

Lorenzo Sadun
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Darren Glass
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Last semester, at the end of my linear algebra class, I had one student approach me and inform me that she wanted to go to graduate school and get a PhD in linear algebra. While I was of course excited for this student and wanted to be encouraging, I did feel a need to explain to her that, while in some sense linear algebra is still an active and vibrant area of research with new applications appearing daily, most of this work bears little resemblance to the solving systems of linear equations and matrix algebra which had taken up the bulk of our semester. I would like to think that I did a good job of explaining to her what comes next in linear algebra, but I wish I had been familiar with Lorenzo Sadun's book Applied Linear Algebra: The Decoupling Principle at the time, because I would have handed it to her immediately to show her a bit more of what linear algebra can be.

Now in its second edition, Sadun's book is intended as a textbook for a second course (or the second and third quarters of a three quarter sequence) in linear algebra. The central theme throughout the book is what Sadun calls the decoupling principle, which is the idea that by simultaneously diagonalizing a system of operators, one can break the system into pieces that are more easily understood. This guiding principle can be used to understand Markov processes, coupled oscillations, Fourier series, difference equations, and any number of other applications, many of which are treated in depth in the book and motivated in an introductory chapter.

Applied Linear Algebra assumes that the reader will have familiarity with matrix operations and row reduction (although even these are covered briefly in an appendix), but little else. The early chapters of this book cover abstract vector spaces, bases, linear transformations, eigenvalues and eigenvectors. He then moves on to consider a number of different applications of these ideas before moving on with the theory.

In the process of doing this, the book has introduced the idea that it is nice when one can find a basis of a vector space made up of eigenvectors for a given linear transformation and it is also nice when one can have an orthogonal basis. The natural question to ask is whether it is possible to have a basis of eigenvectors which are also orthogonal. As most readers of this review likely know, the answer is "sometimes," and to fully understand this answer involves understanding adjoints and Hermitian operators and things of that nature, which Sadun covers in depth in the middle chapters of his book.

A final set of chapters apply all of these tools to look at applications, primarily from physics, such as the wave equation, Fourier transforms, and Green's functions. Along the way, Sadun explores the similarities and differences between the finite dimensional and infinite dimensional situations.

This may sound like a lot of material, but Sadun's writing style makes it all seem very natural and straightforward. The book has a large number of exercises, ranging from the computational to the theoretical. Many sections also have what the author calls an 'exploration', which is a series of questions leading the reader through a topic such as "Curve Fitting", "The Central Limit Theorem", or "Discreting the Wave Equation." There are numerous examples worked out throughout the book, and many of the exercises come with hints and solutions.

For all of these reasons, I can imagine using Sadun's book in a number of different ways. At our small college we only teach one semester of linear algebra, but Sadun's exposition is so clear and littered with examples and motivation for the material that I would not hesitate to point a student wishing to do an independent study to this book. If we were to teach a second semester of linear algebra, I would certainly consider Sadun's book to be a frontrunner as a choice of texts. As it is, I will keep this book on my desk the next time I teach our existing linear algebra course, as a source of examples, problems, and ideas for my own teaching.

Darren Glass is an Assistant Professor of Mathematics at Gettysburg College. He can be reached at