Applied Linear Algebra The Decoupling Principle

Preface The purpose of the book This book was designed as a textbook for a junior-senior level second course in linear algebra at the University of Texas at Austin. In that course, and in this book, I try to show math, physics, and engineering majors the incredible power of linear algebra in the real world. The hope is that, when faced with a linear system (or a nonlinear system that can be reasonably linearized), future engineers will think to decompose the system into modes that they can understand. Usually this is done by diagonalization. Sometimes this is done by decomposing into a convenient orthonormal basis, such as Fourier series. Sometimes a continuous decomposition, into d functions or by Fourier transforms, is called for. The underlying ideas of breaking a vector into modes (the Superposition Principle) and of decoupling a complicated system by a suitable choice of linear coordinates (the Decoupling Principle) appear throughout physics and engineering. My goal is to impress upon students the importance of these principles, while giving them enough tools to use them effectively. There are many existing types of second linear algebra courses, and many books to match, but few if any make this goal a priority. Some courses are theoretical, going in the direction of functional analysis, Lie Groups or abstract algebra. "Applied" second courses tend to be heavily numerical, teaching efficient and robust algorithms for factorizing or diagonalizing matrices. Some courses split the difference, developing matrix theory in depth, proving classification theorems (e.g., Jordan form) and estimates (e.g., Gershegorin's Theorem). While each of these courses is well-suited for its chosen audience, none give a prospective physicist or engineer substantial insight into how or why to apply linear algebra at all. Notes to the instructor The readers of this book are assumed to have taken an introductory linear algebra class, and hence to be familiar with basic matrix operations such as row reduction, matrix multiplication and inversion, and taking determinants. The reader is also assumed to have had some exposure to vector spaces and linear transformations. This material is reviewed in Chapters 2 and 3, pretty much from the beginning, but a student who has never seen an abstract vector space will have trouble keeping up. The subject of Chapter 4, eigenvalues, is typically covered quite hastily at the end of a first course (if at all), so I work under the assumption that readers do not have any prior knowledge of eigenvalues. The key concept of these introductory chapters is that a basis makes a vector space look likeR n (or sometimesC n ) and makes linear transformations look like matrices. Some bases make the conversion process simple, while others make the end results simple. The standard basis inR n makes coordinates easy to find, but may result in an operator being represented by an ugly matrix. A basis of eigenvectors, on the other hand, makes the operator appear simple but makes finding the coordinates of a vector difficult. To handle problems in linear algebra, one must be adept in coordinatization and in performing change-of-basis operations, both for vectors and for operators. One premise of this book is that standard software packages (e.g., MATLAB, Maple or Mathematica) make it easy to diagonalize matrices without any knowledge of sophisticated numerical algorithms. This frees us to consider theuseof diagonalization, and some general features of important classes of operators (e.g., Hermitian or unitary operators). Diagonalization, by computer or by hand, gives a set of coordinates in which a problem, even a problem with an infinite number of degrees of freedom, decouples into a collection of independent scalar equations. This is what I call the Decoupling Principle. (Strictly sSadun, Lorenzo A. is the author of 'Applied Linear Algebra The Decoupling Principle', published 2000 under ISBN 9780130856456 and ISBN 0130856452.