Applied Linear Algebra

by Olver, Peter J.; Shakiban, Cheri

• ISBN13: 9780131473829
• ISBN10: 0131473824
• Edition: 1st
• Format: Paperback
• Copyright: 2005-01-10
• Publisher: Pearson

Summary

This book describes basic methods and algorithms used in modern, real problems likely to be encountered by engineers and scientists - and fosters an understanding of why mathematical techniques work and how they can be derived from first principles.Assumes no previous exposure to linear algebra. Presents applications hand in hand with theory, leading readers through the reasoning that leads to the important results. Provides theorems and proofs where needed. Features abundant exercises after almost every subsection, in a wide range of difficulty.A thorough reference for engineers and scientists.

Table of Contents

Preface	xi
Linear Algebraic Systems	1	(76)
Solution of Linear Systems	2	(2)
Matrices and Vectors	4	(8)
Matrix Arithmetic	6	(6)
Gaussian Elimination---Regular Case	12	(10)
Elementary Matrices	16	(2)
The L U Factorization	18	(2)
Forward and Back Substitution	20	(2)
Pivoting and Permutations	22	(9)
Permutation Matrices	25	(2)
The Permuted L U Factorization	27	(4)
Matrix Inverses	31	(13)
Gauss--Jordan Elimination	36	(5)
Solving Linear Systems with the Inverse	41	(1)
The L D V Factorization	42	(2)
Transposes and Symmetric Matrices	44	(5)
Factorization of Symmetric Matrices	46	(3)
Practical Linear Algebra	49	(11)
Tridiagonal Matrices	53	(3)
Pivoting Strategies	56	(4)
General Linear Systems	60	(10)
Homogeneous Systems	68	(2)
Determinants	70	(7)
Vector Spaces and Bases	77	(53)
Real Vector Spaces	78	(5)
Subspaces	83	(6)
Span and Linear Independence	89	(11)
Linear Independence and Dependence	94	(6)
Bases and Dimension	100	(7)
The Fundamental Matrix Subspaces	107	(15)
Kernel and Range	107	(4)
The Superposition Principle	111	(3)
Adjoint Systems, Cokernel, and Corange	114	(2)
The Fundamental Theorem of Linear Algebra	116	(6)
Graphs and Incidence Matrices	122	(8)
Inner Products and Norms	130	(50)
Inner Products	131	(6)
Inner Products on Function Spaces	134	(3)
Inequalities	137	(7)
The Cauchy--Schwarz Inequality	138	(2)
Orthogonal Vectors	140	(2)
The Triangle Inequality	142	(2)
Norms	144	(9)
Unit Vectors	147	(3)
Equivalence of Norms	150	(3)
Positive Definite Matrices	153	(10)
Gram Matrices	157	(6)
Completing the Square	163	(6)
The Cholesky Factorization	168	(1)
Complex Vector Spaces	169	(11)
Complex Numbers	170	(4)
Complex Vector Spaces and Inner Products	174	(6)
Minimization and Least Squares Approximation	180	(37)
Minimization Problems	180	(4)
Equilibrium Mechanics	181	(1)
Solution of Equations	181	(2)
The Closest Point	183	(1)
Minimization of Quadratic Functions	184	(5)
Least Squares and the Closest Point	189	(7)
Least Squares	193	(3)
Data Fitting and Interpolation	196	(21)
Polynomial Approximation and Interpolation	201	(8)
Approximation and Interpolation by General Functions	209	(2)
Weighted Least Squares	211	(2)
Least Squares Approximation in Function Spaces	213	(4)
Orthogonality	217	(76)
Orthogonal Bases	218	(9)
Computations in Orthogonal Bases	222	(5)
The Gram--Schmidt Process	227	(8)
Modifications of the Gram--Schmidt Process	232	(3)
Orthogonal Matrices	235	(13)
The Q R Factorization	240	(3)
III-Conditioned Systems and Householder's Method	243	(5)
Orthogonal Polynomials	248	(8)
The Legendre Polynomials	249	(4)
Other Systems of Orthogonal Polynomials	253	(3)
Orthogonal Projections and Least Squares	256	(12)
Orthogonal Projection	256	(4)
Orthogonal Least Squares	260	(5)
Orthogonal Polynomials and Least Squares	265	(3)
Orthogonal Subspaces	268	(9)
Orthogonality of the Fundamental Matrix Subspaces and the Fredholm Alternative	272	(5)
Discrete Fourier Series and the Fast Fourier Transform	277	(16)
Compression and Noise Removal	285	(2)
The Fast Fourier Transform	287	(6)
Equilibrium	293	(37)
Springs and Masses	294	(7)
Positive Definiteness and the Minimization Principle	299	(2)
Electrical Networks	301	(11)
Batteries, Power, and the Electrical--Mechanical Correspondence	307	(5)
Structures	312	(18)
Linearity	330	(60)
Linear Functions	331	(15)
Linear Operators	336	(2)
The Space of Linear Functions	338	(3)
Composition	341	(3)
Inverses	344	(2)
Linear Transformations	346	(12)
Change of Basis	353	(5)
Affine Transformations and Isometries	358	(5)
Isometry	360	(3)
Linear Systems	363	(19)
The Superposition Principle	365	(5)
Inhomogeneous Systems	370	(5)
Superposition Principles for Inhomogeneous Systems	375	(2)
Complex Solutions to Real Systems	377	(5)
Adjoints	382	(8)
Self--Adjoint and Positive Definite Linear Functions	385	(2)
Minimization	387	(3)
Eigenvalues	390	(55)
Simple Dynamical Systems	391	(4)
Scalar Ordinary Differential Equations	391	(3)
First Order Dynamical Systems	394	(1)
Eigenvalues and Eigenvectors	395	(11)
Basic Properties of Eigenvalues	401	(5)
Eigenvector Bases and Diagonalization	406	(7)
Diagonalization	409	(4)
Eigenvalues of Symmetric Matrices	413	(12)
The Spectral Theorem	418	(4)
Optimization Principles for Eigenvalues	422	(3)
Singular Values	425	(9)
Condition Number, Rank, and Principal Component Analysis	429	(1)
The Pseudoinverse	430	(4)
Incomplete Matrices	434	(11)
The Schur Decomposition	435	(3)
The Jordan Canonical Form	438	(7)
Linear Dynamical Systems	445	(65)
Basic Solution Techniques	446	(13)
The Phase Plane	447	(3)
Existence and Uniqueness	450	(2)
Complete Systems	452	(4)
The General Case	456	(3)
Stability of Linear Systems	459	(6)
Two-Dimensional Systems	465	(7)
Distinct Real Eigenvalues	466	(2)
Complex Conjugate Eigenvalues	468	(1)
Incomplete Double Real Eigenvalue	469	(1)
Complete Double Real Eigenvalue	469	(3)
Matrix Exponentials	472	(13)
Inhomogeneous Linear Systems	478	(3)
Applications in Geometry	481	(4)
Dynamics of Structures	485	(15)
Stable Structures	487	(5)
Unstable Structures	492	(3)
Systems with Differing Masses	495	(2)
Friction and Damping	497	(3)
Forcing and Resonance	500	(10)
Electrical Circuits	504	(2)
Forcing and Resonance in Systems	506	(4)
Iteration of Linear Systems	510	(73)
Linear Iterative Systems	511	(12)
Scalar Systems	511	(3)
Powers of Matrices	514	(6)
Diagonalization and Iteration	520	(3)
Stability	523	(7)
Fixed Points	528	(2)
Matrix Norms and the Gerschgorin Theorem	530	(10)
Matrix Norms	530	(2)
Explicit Formulae	532	(4)
The Gerschgorin Circle Theorem	536	(4)
Markov Processes	540	(6)
Iterative Solution of Linear Algebraic Systems	546	(22)
The Jacobi Method	549	(3)
The Gauss--Seidel Method	552	(5)
Successive Over-Relaxation (SOR)	557	(5)
Conjugate Gradients	562	(6)
Numerical Computation of Eigenvalues	568	(15)
The Power Method	568	(4)
The Q R Algorithm	572	(6)
Tridiagonalization	578	(5)
Boundary Value Problems in One Dimension	583	(67)
Elastic Bars	584	(9)
Generalized Functions and the Green's Function	593	(19)
The Delta Function	594	(4)
Calculus of Generalized Functions	598	(8)
The Green's Function	606	(6)
Adjoints and Minimum Principles	612	(11)
Adjoints of Differential Operators	612	(4)
Positivity and Minimum Principles	616	(4)
Inhomogeneous Boundary Conditions	620	(3)
Beams and Splines	623	(12)
Splines	630	(5)
Sturm--Liouville Boundary Value Problems	635	(5)
Finite Elements	640	(10)
References	650	(3)
Answers to Selected Exercises	653	(46)
Index	699