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Applied linear algebra
,
Book
Olver, Peter J
;
Shakiban, Chehrzad
Springer International Publishing AG
2018
Cataloging brief
Applied linear algebra
,
Book
Olver, Peter J
;
Shakiban, Chehrzad
Springer International Publishing AG
2018
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Preface
(7)
Syllabi and Prerequisites
(9)
Survey of Topics
(10)
Course Outlines
(12)
Comments on Individual Chapters
(14)
Changes from the First Edition
(15)
Exercises and Software
(16)
Conventions and Notations
(17)
History and Biography
(19)
Some Final Remarks
(19)
Acknowledgments
(20)
Table of Contents
(21)
Chapter 1: Linear Algebraic Systems
(26)
1.1 Solution of Linear Systems
(26)
1.2 Matrices and Vectors
(28)
Matrix Arithmetic
(30)
1.3 Gaussian Elimination—Regular Case
(37)
Elementary Matrices
(41)
The LU Factorization
(43)
Forward and Back Substitution
(45)
1.4 Pivoting and Permutations
(47)
Permutations and Permutation Matrices
(50)
The Permuted LU Factorization
(52)
1.5 Matrix Inverses
(56)
Gauss–Jordan Elimination
(60)
Solving Linear Systems with the Inverse
(65)
The LDV Factorization
(66)
1.6 Transposes and Symmetric Matrices
(68)
Factorization of Symmetric Matrices
(70)
1.7 Practical Linear Algebra
(73)
Tridiagonal Matrices
(77)
Pivoting Strategies
(80)
1.8 General Linear Systems
(84)
Homogeneous Systems
(92)
1.9 Determinants
(94)
Chapter 2: Vector Spaces and Bases
(100)
2.1 Real Vector Spaces
(101)
2.2 Subspaces
(106)
2.3 Span and Linear Independence
(112)
Linear Independence and Dependence
(117)
2.4 Basis and Dimension
(123)
2.5 The Fundamental Matrix Subspaces
(130)
Kernel and Image
(130)
The Superposition Principle
(135)
Adjoint Systems, Cokernel, and Coimage
(137)
The Fundamental Theorem of Linear Algebra
(139)
2.6 Graphs and Digraphs
(145)
Chapter 3: Inner Products and Norms
(154)
3.1 Inner Products
(154)
Inner Products on Function Spaces
(158)
3.2 Inequalities
(162)
The Cauchy–Schwarz Inequality
(162)
Orthogonal Vectors
(165)
The Triangle Inequality
(167)
3.3 Norms
(169)
Unit Vectors
(173)
Equivalence of Norms
(175)
Matrix Norms
(178)
3.4 Positive Definite Matrices
(181)
Gram Matrices
(186)
3.5 Completing the Square
(191)
The Cholesky Factorization
(196)
3.6 Complex Vector Spaces
(197)
Complex Numbers
(198)
Complex Vector Spaces and Inner Products
(202)
Chapter 4: Orthogonality
(208)
4.1 Orthogonal and Orthonormal Bases
(209)
Computations in Orthogonal Bases
(213)
4.2 The Gram–Schmidt Process
(217)
Modifications of the Gram–Schmidt Process
(222)
4.3 Orthogonal Matrices
(225)
The QR Factorization
(230)
Ill-Conditioned Systems and Householder’s Method
(233)
4.4 Orthogonal Projections and Orthogonal Subspaces
(237)
Orthogonal Projection
(238)
Orthogonal Subspaces
(241)
Orthogonality of the Fundamental Matrix Subspaces and the Fredholm Alternative
(246)
4.5 Orthogonal Polynomials
(251)
The Legendre Polynomials
(252)
Other Systems of Orthogonal Polynomials
(256)
Chapter 5: Minimization and Least Squares
(260)
5.1 Minimization Problems
(260)
Equilibrium Mechanics
(261)
Solution of Equations
(261)
The Closest Point
(263)
5.2 Minimization of Quadratic Functions
(264)
5.3 The Closest Point
(270)
5.4 Least Squares
(275)
5.5 Data Fitting and Interpolation
(279)
Polynomial Approximation and Interpolation
(284)
Approximation and Interpolation by General Functions
(296)
Least Squares Approximation in Function Spaces
(299)
Orthogonal Polynomials and Least Squares
(302)
Splines
(304)
5.6 Discrete Fourier Analysis and the Fast Fourier Transform
(310)
Compression and Denoising
(318)
The Fast Fourier Transform
(320)
Chapter 6: Equilibrium
(326)
6.1 Springs and Masses
(326)
Positive Definiteness and the Minimization Principle
(334)
6.2 Electrical Networks
(336)
Batteries, Power, and the Electrical–Mechanical Correspondence
(342)
6.3 Structures
(347)
Chapter 7: Linearity
(366)
7.1 Linear Functions
(367)
Linear Operators
(372)
The Space of Linear Functions
(374)
Dual Spaces
(375)
Composition
(377)
Inverses
(380)
7.2 Linear Transformations
(383)
Change of Basis
(390)
7.3 Affine Transformations and Isometries
(395)
Isometry
(397)
7.4 Linear Systems
(401)
The Superposition Principle
(403)
Inhomogeneous Systems
(408)
Superposition Principles for Inhomogeneous Systems
(413)
Complex Solutions to Real Systems
(415)
7.5 Adjoints, Positive Definite Operators, and Minimization Principles
(420)
Self-Adjoint and Positive Definite Linear Functions
(423)
Minimization
(425)
Chapter 8: Eigenvalues and Singular Values
(428)
8.1 Linear Dynamical Systems
(429)
Scalar Ordinary Differential Equations
(429)
First Order Dynamical Systems
(432)
8.2 Eigenvalues and Eigenvectors
(433)
Basic Properties of Eigenvalues
(440)
The Gershgorin Circle Theorem
(445)
8.3 Eigenvector Bases
(448)
Diagonalization
(451)
8.4 Invariant Subspaces
(454)
8.5 Eigenvalues of Symmetric Matrices
(456)
The Spectral Theorem
(462)
Optimization Principles for Eigenvalues of Symmetric Matrices
(465)
8.6 Incomplete Matrices
(469)
The Schur Decomposition
(469)
The Jordan Canonical Form
(472)
8.7 Singular Values
(479)
The Pseudoinverse
(482)
The Euclidean Matrix Norm
(484)
Condition Number and Rank
(485)
Spectral Graph Theory
(487)
8.8 Principal Component Analysis
(492)
Variance and Covariance
(492)
The Principal Components
(496)
Chapter 9: Iteration
(500)
9.1 Linear Iterative Systems
(501)
Scalar Systems
(501)
Powers of Matrices
(504)
Diagonalization and Iteration
(509)
9.2 Stability
(513)
Spectral Radius
(514)
Fixed Points
(518)
Matrix Norms and Convergence
(520)
9.3 Markov Processes
(524)
9.4 Iterative Solution of Linear Algebraic Systems
(531)
The Jacobi Method
(533)
The Gauss–Seidel Method
(537)
Successive Over-Relaxation
(542)
9.5 Numerical Computation of Eigenvalues
(547)
The Power Method
(547)
The QR Algorithm
(551)
Tridiagonalization
(557)
9.6 Krylov Subspace Methods
(561)
Krylov Subspaces
(561)
Arnoldi Iteration
(562)
The Full Orthogonalization Method
(565)
The Conjugate Gradient Method
(567)
The Generalized Minimal Residual Method
(571)
9.7 Wavelets
(574)
The Haar Wavelets
(574)
Modern Wavelets
(580)
Solving the Dilation Equation
(584)
Chapter 10: Dynamics
(589)
10.1 Basic Solution Techniques
(589)
The Phase Plane
(591)
Existence and Uniqueness
(594)
Complete Systems
(596)
The General Case
(599)
10.2 Stability of Linear Systems
(603)
10.3 Two-Dimensional Systems
(609)
Distinct Real Eigenvalues
(610)
Complex Conjugate Eigenvalues
(611)
Incomplete Double Real Eigenvalue
(612)
Complete Double Real Eigenvalue
(612)
10.4 Matrix Exponentials
(616)
Applications in Geometry
(623)
Invariant Subspaces and Linear Dynamical Systems
(627)
Inhomogeneous Linear Systems
(629)
10.5 Dynamics of Structures
(632)
Stable Structures
(634)
Unstable Structures
(639)
Systems with Differing Masses
(642)
Friction and Damping
(644)
10.6 Forcing and Resonance
(647)
Electrical Circuits
(652)
Forcing and Resonance in Systems
(654)
References
(657)
Symbol Index
(661)
Subject Index
(666)