内容简介
Linear algebra permeates
mathematics, perhaps
more so than any other
single subject. It plays an
essential role in pure and
applied mathematics,
statistics, computer
science, and many aspects
of physics and
engineering. This book
conveys in a user-friendly
way the basic and
advanced techniques of
linear algebra from the
point of view of a working
analyst. The techniques
are illustrated by a wide
sample of applications and
examples that are chosen
to highlight the tools of the
trade. In short, this is
material that many of us
wish we had been taught
as graduate
students.Roughly the first
third of the book covers
the basic material of a first
course in linear algebra.
The remaining chapters
are devoted to applications
drawn from vector
calculus, numerical
analysis, control theory,
complex analysis, convexity
and functional analysis. In
particular, fixed point
theorems, extremal
problems, matrix
equations, zero location
and eigenvalue location
problems, and matrices
with nonnegative entries
are discussed. Appendices
on useful facts from
analysis and
supplementary information
from complex function
theory are also provided
for the convenience of the
reader.In this new edition,
most of the chapters in the
first edition have been
revised, some extensively.
The revisions include
changes in a number of
proofs, either to simplify
the argument, to make the
logic clearer or, on
occasion, to sharpen the
result. New introductory
sections on linear
programming, extreme
points for polyhedra and a
Nevanlinna-Pick
interpolation problem have
been added, as have some
very short introductory
sections on the
mathematics behind
Google, Drazin inverses,
band inverses and
applications of SVD
together with a number of
new exercises.
目录
Preface to the Second Edition
Preface to the First Edition
Chapter 1.Vector spaces
§1.1.Preview
§1.2.The abstract definition of a vector space
§1.3.Some definitions
§1.4.Mappings
§1.5.Triangular matrices
§1.6.Block triangular matrices
§1.7.Schur complements
§1.8.Other matrix products
Chapter 2.Gaussian elimination
§2.1.Some preliminary observations
§2.2.Examples
§2.3.Upper echelon matrices
§2.4.The conservation of dimension
§2.5.Quotient spaces
§2.6.Conservation of dimension for matrices
§2.7.From U to A
§2.8.Square matrices
Chapter 3.Additional applications of Gaussian elimination
§3.1.Gaussian elimination redux
§3.2.Properties of BA and AC
§3.3.Extracting a basis
§3.4.Computing the coefficients in a basis
§3.5.The Gauss-Seidel method
§3.6.Block Gaussian elimination
§3.7.{0, 1, c§}
§3.8.Review
Chapter 4.Eigenvalues and eigenvectors
§4.1.Change of basis and similarity
§4.2.Invariant subspaces
§4.3.Existence of eigenvalues
§4.4.Eigenvalues for matrices
§4.5.Direct sums
§4.6.Diagonalizable matrices
§4.7.An algorithm for diagonalizing matrices
§4.8.Computing eigenvalues at this point
§4.9.Not all matrices are diagonalizable
§4.10.The Jordan decomposition theorem
§4.11.An instructive example
§4.12.The binomial formula
§4.13.More direct sum decompositions
§4.14.Verification of Theorem 4.13
§4.15.Bibliographical notes
Chapter 5.Determinants
§5.1.Functionals
§5.2.Determinants
§5.3.Useful rules for calculating determinants
§5.4.Eigenvalues
