Page 6 - Matrix Analysis & Applied Linear Algebra

x Preface

before eigenvalues or singular values are thoroughly discussed. To compensate,
I try to provide enough “wiggle room” so that an instructor can temper the
inefficiencies by tailoring the approach to the students’ prior background.

Comprehensiveness and Flexibility
A rather comprehensive treatment of linear algebra and its applications is
presented and, consequently, the book is not meant to be devoured cover-to-cover
in a typical one-semester course. However, the presentation is structured to pro-
vide flexibility in topic selection so that the text can be easily adapted to meet
the demands of different course outlines without suffering breaks in continuity.
Each section contains basic material paired with straightforward explanations,
examples, and exercises. But every section also contains a degree of depth coupled
with thought-provoking examples and exercises that can take interested students
to a higher level. The exercises are formulated not only to make a student think
about material from a current section, but they are designed also to pave the way
for ideas in future sections in a smooth and often transparent manner. The text
accommodates a variety of presentation levels by allowing instructors to select
sections, discussions, examples, and exercises of appropriate sophistication. For
example, traditional one-semester undergraduate courses can be taught from the
basic material in Chapter 1 (Linear Equations); Chapter 2 (Rectangular Systems
and Echelon Forms); Chapter 3 (MatrixAlgebra); Chapter 4 (Vector Spaces);
Chapter 5 (Norms, Inner Products, and Orthogonality); Chapter 6 (Determi-
nants); and Chapter 7 (Eigenvalues and Eigenvectors). The level of the course
and the degree of rigor are controlled by the selection and depth of coverage in
the latter sections of Chapters 4, 5, and 7. An upper-level course might consist
of a quick review of Chapters 1, 2, and 3 followed by a more in-depth treatment
of Chapters 4, 5, and 7. For courses containing advanced undergraduate or grad-
uate students, the focus can be on material in the latter sections of Chapters 4,
5, 7, and Chapter 8 (Perron–Frobenius Theory of Nonnegative Matrices). A rich
two-semester course can be taught by using the text in its entirety.

What Does “Applied” Mean?
Most people agree that linear algebra is at the heart of applied science, but
there are divergent views concerning what “applied linear algebra” really means;
the academician’s perspective is not always the same as that of the practitioner.
In a poll conducted by SIAM in preparation for one of the triannual SIAM con-
ferences on applied linear algebra, a diverse group of internationally recognized
scientific corporations and government laboratories was asked how linear algebra
finds application in their missions. The overwhelming response was that the pri-
mary use of linear algebra in applied industrial and laboratory work involves the
development, analysis, and implementation of numerical algorithms along with
some discrete and statistical modeling. The applications in this book tend to
reflect this realization. While most of the popular “academic” applications are
included, and “applications” to other areas of mathematics are honestly treated,