Matrix Analysis and Applied Linear Algebra

by Meyer, Carl D.

• ISBN13: 9780898714548
• ISBN10: 0898714540
• Format: Paperback
• Copyright: 2000-06-01
• Publisher: SIAM

Summary

This book avoids the traditional definition-theorem-proof format; instead a fresh approach introduces a variety of problems and examples all in a clear and informal style. The in-depth focus on applications separates this book from others, and helps students to see how linear algebra can be applied to real-life situations. Some of the more contemporary topics of applied linear algebra are included here which are not normally found in undergraduate textbooks. Theoretical developments are always accompanied with detailed examples, and each section ends with a number of exercises from which students can gain further insight. Moreover, the inclusion of historical information provides personal insights into the mathematicians who developed this subject. The textbook contains numerous examples and exercises, historical notes, and comments on numerical performance and the possible pitfalls of algorithms. Solutions to all of the exercises are provided, as well as a CD-ROM containing a searchable copy of the textbook.

Author Biography

Carl Meyer is a Professor in the Department of Mathematics at North Carolina State University

Table of Contents

Preface	ix
Linear Equations	1	(40)
Introduction	1	(2)
Gaussian Elimination and Matrices	3	(12)
Gauss--Jordan Method	15	(3)
Two-Point Boundary Value Problems	18	(3)
Making Gaussian Elimination Work	21	(12)
III-Conditioned Systems	33	(8)
Rectangular Systems and Echelon Forms	41	(38)
Row Echelon Form and Rank	41	(6)
Reduced Row Echelon Form	47	(6)
Consistency of Linear Systems	53	(4)
Homogeneous Systems	57	(7)
Nonhomogeneous Systems	64	(9)
Electrical Circuits	73	(6)
Matrix Algebra	79	(80)
From Ancient China to Arthur Cayley	79	(2)
Addition and Transposition	81	(8)
Linearity	89	(4)
Why Do it this Way	93	(2)
Matrix Multiplication	95	(10)
Properties of Matrix Multiplication	105	(10)
Matrix Inversion	115	(9)
Inverses of Sums and Sensitivity	124	(7)
Elementary Matrices and Equivalence	131	(10)
The LU Factorization	141	(18)
Vector Spaces	159	(110)
Spaces and Subspaces	159	(10)
Four Fundamental Subspaces	169	(12)
Linear Independence	181	(13)
Basis and Dimension	194	(16)
More about Rank	210	(13)
Classical Least Squares	223	(15)
Linear Transformations	238	(13)
Change of Basis and Similarity	251	(8)
Invariant Subspaces	259	(10)
Norms, Inner Products, and Orthogonality	269	(190)
Vector Norms	269	(10)
Matrix Norms	279	(7)
Inner-Product Spaces	286	(8)
Orthogonal Vectors	294	(13)
Gram--Schmidt Procedure	307	(13)
Unitary and Orthogonal Matrices	320	(21)
Orthogonal Reduction	341	(15)
Discrete Fourier Transform	356	(27)
Complementary Subspaces	383	(11)
Range-Nullspace Decomposition	394	(9)
Orthogonal Decomposition	403	(8)
Singular Value Decomposition	411	(18)
Orthogonal Projection	429	(17)
Why Least Squares?	446	(4)
Angles between Subspaces	450	(9)
Determinants	459	(30)
Determinants	459	(16)
Additional Properties of Determinants	475	(14)
Eigenvalues and Eigenvectors	489	(172)
Elementary Properties of Eigensystems	489	(16)
Diagonalization by Similarity Transformations	505	(20)
Functions of Diagonalizable Matrices	525	(16)
Systems of Differential Equations	541	(6)
Normal Matrices	547	(11)
Positive Definite Matrices	558	(16)
Nilpotent Matrices and Jordan Structure	574	(13)
Jordan Form	587	(12)
Functions of Nondiagonalizable Matrices	599	(17)
Difference Equations, Limits, and Summability	616	(26)
Minimum Polynomials and Krylov Methods	642	(19)
Perron--Frobenius Theory	661	(44)
Introduction	661	(2)
Positive Matrices	663	(7)
Nonnegative Matrices	670	(17)
Stochastic Matrices and Markov Chains	687	(18)
Index	705