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1.

図書

図書
David C. Lay
出版情報: Reading, Mass. : Addison-Wesley Pub. Co., c2000  xx, 486, 57, 12 p. ; 24 cm
シリーズ名: Addison-Wesley world student series
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Linear Equations in Linear Algebra / 1:
Introductory Example: Linear Models in Economics and Engineering
Systems of Linear Equations / 1.1:
Row Reduction and Echelon Forms / 1.2:
Vector Equations / 1.3:
The Matrix Equation Ax = b / 1.4:
Solution Sets of Linear Systems / 1.5:
Applications of Linear Systems / 1.6:
Linear Independence / 1.7:
Introduction to Linear Transformations / 1.8:
The Matrix of a Linear Transformation / 1.9:
Linear Models in Business, Science, and Engineering / 1.10:
Supplementary Exercises
Matrix Algebra / 2:
Introductory Example: Computer Models in Aircraft Design
Matrix Operations / 2.1:
The Inverse of a Matrix / 2.2:
Characterizations of Invertible Matrices / 2.3:
Partitioned Matrices / 2.4:
Matrix Factorizations / 2.5:
The Leontief Input=Output Model / 2.6:
Applications to Computer Graphics / 2.7:
Subspaces of Rn / 2.8:
Dimension and Rank / 2.9:
Determinants / 3:
Introductory Example: Determinants in Analytic Geometry
Introduction to Determinants / 3.1:
Properties of Determinants / 3.2:
Cramer's Rule, Volume, and Linear Transformations / 3.3:
Vector Spaces / 4:
Introductory Example: Space Flight and Control Systems
Vector Spaces and Subspaces / 4.1:
Null Spaces, Column Spaces, and Linear Transformations / 4.2:
Linearly Independent Sets Bases / 4.3:
Coordinate Systems / 4.4:
The Dimension of a Vector Space / 4.5:
Rank / 4.6:
Change of Basis / 4.7:
Applications to Difference Equations / 4.8:
Applications to Markov Chains / 4.9:
Eigenvalues and Eigenvectors / 5:
Introductory Example: Dynamical Systems and Spotted Owls
Eigenvectors and Eigenvalues / 5.1:
The Characteristic Equation / 5.2:
Diagonalization / 5.3:
Eigenvectors and Linear Transformations / 5.4:
Complex Eigenvalues / 5.5:
Discrete Dynamical Systems / 5.6:
Applications to Differential Equations / 5.7:
Iterative Estimates for Eigenvalues / 5.8:
Orthogonality and Least Squares / 6:
Introductory Example: Readjusting the North American Datum
Inner Product, Length, and Orthogonality / 6.1:
Orthogonal Sets / 6.2:
Orthogonal Projections / 6.3:
The Gram-Schmidt Process / 6.4:
Least-Squares Problems / 6.5:
Applications to Linear Models / 6.6:
Inner Product Spaces / 6.7:
Applications of Inner Product Spaces / 6.8:
Symmetric Matrices and Quadratic Forms / Chapter 7:
Introductory Example: Multichannel Image Processing
Diagonalization of Symmetric Matrices / 7.1:
Quadratic Forms / 7.2:
Constrained Optimization / 7.3:
The Singular Value Decomposition / 7.4:
Applications to Image Processing and Statistics / 7.5:
Supplementary Exercises (ONLINE ONLY)
The Geometry of Vector Spaces / 8:
Introductory Example: The Platonic Solids
Affine Combinations / 8.1:
Affine Independence / 8.2:
Convex Combinations / 8.3:
Hyperplanes / 8.4:
Polytopes / 8.5:
Curves and Surfaces / 8.6:
Optimization / 9:
Introductory Example: The Berlin Airlift
Matrix Games / 9.1:
Linear Programming - Geometric Method / 9.2:
Linear Programming - Simplex Method / 9.3:
Duality / 9.4:
Appendices
Uniqueness of the Reduced Echelon Form / A:
Complex Numbers / B:
Glossary
Answers to Odd-Numbered Exercises
Index
Linear Equations in Linear Algebra / 1:
Introductory Example: Linear Models in Economics and Engineering
Systems of Linear Equations / 1.1:
2.

図書

図書
Alexander Kleshchev
出版情報: Cambridge : Cambridge University Press, c2005  xiv, 277 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 163
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Preface
Linear Representations / Part I:
Notations and generalities / 1:
Symmetric groups I / 2:
Degenerate affine Hecke algebra / 3:
First results on Hn-modules / 4:
Crystal operators / 5:
Character calculations / 6:
Integral representations and cyclotomic Hecke algebras / 7:
Functors ei? and fi? / 8:
Construction of UZ+ and irreducible modules / 9:
Identification of the crystal / 10:
Symmetric groups II / 11:
Projective Representations / Part II:
Generalities on superalgebra / 12:
Sergeev superalgebras / 13:
Affine Sergeev superalgebras / 14:
Integral representations and cyclotomic Sergeev algebras / 15:
First results on Xn-modules / 16:
Crystal operators for Xn / 17:
Character calculations for Xn / 18:
Operators ei? and fi? / 19:
Double covers / 20:
References
Index
Preface
Linear Representations / Part I:
Notations and generalities / 1:
3.

図書

図書
Richard S. Varga
出版情報: Berlin : Springer, c2004  x, 226 p. ; 24 cm
シリーズ名: Springer series in computational mathematics ; 36
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Preface / I:
Basic Theory / 1:
Geršgorin's Theorem / 1.1:
Extensions of Geršgorin's Theorem via Graph Theory / 1.2:
Analysis Extensions of Geršgorin's Theorem and Fan's Theorem / 1.3:
A Norm Derivation of Geršgorin's Theorem 1.1 / 1.4:
Geršgorin-Type Eigenvalue Inclusion Theorems / 2:
Brauer's Ovals of Cassini / 2.1:
Higher-Order Lemniscates / 2.2:
Comparison of the Brauer Sets and the Brualdi Sets / 2.3:
The Sharpness of Brualdi Lemniscate Sets / 2.4:
An Example / 2.5:
More Eigenvalue Inclusion Results / 3:
The Parodi-Schneider Eigenvalue Inclusion Sets / 3.1:
The Field of Values of a Matrix / 3.2:
Newer Eigenvalue Inclusion Sets / 3.3:
The Pupkov-Solov'ev Eigenvalue Inclusions Set / 3.4:
Minimal Geršgorin Sets and Their Sharpness / 4:
Minimal Geršgorin Sets / 4.1:
Minimal Geršgorin Sets via Permutations / 4.2:
A Comparison of Minimal Geršgorin Sets and Brualdi Sets / 4.3:
G-Functions / 5:
The Sets <$>{\cal F}_{\bf n}<$> and <$>{\cal G}_{\bf n}<$> / 5.1:
Structural Properties of <$>{\cal G}_{\bf n}<$> and <$>{\cal G}_{\bf n}^{\bf c}<$> / 5.2:
Minimal G-Functions / 5.3:
Minimal G-Functions with Small Domains of Dependence / 5.4:
Connections with Brauer Sets and Generalized Brualdi Sets / 5.5:
Geršgorin-Type Theorems for Partitioned Matrices / 6:
Partitioned Matrices and Block Diagonal Dominance / 6.1:
A Different Norm Approach / 6.2:
A Variation on a Theme by Brualdi / 6.3:
G-Functions in the Partitioned Case / 6.4:
Geršgorin's Paper from 1931, and Comments / Appendix A:
Vector Norms and Induced Operator Norms / Appendix B:
The Perron-Frobenius Theory of Nonnegative Matrices / Appendix C:
Matlab 6 Programs / Appendix D:
References
Preface / I:
Basic Theory / 1:
Geršgorin's Theorem / 1.1:
4.

図書

図書
James J. Callahan
出版情報: New York : Springer, c2000  xvi, 451 p. ; 25 cm
シリーズ名: Undergraduate texts in mathematics
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Relativity before 1905
Special relativityùkinematics
Special relativityùkinetics
Arbitrary frames
Surfaces and curvatures
Intrinsic geometry
General relativity
Consequences
Relativity before 1905
Special relativityùkinematics
Special relativityùkinetics
5.

図書

図書
Yitzhak Katznelson, Yonatan R. Katznelson
出版情報: Providence, R.I. : American Mathematical Society, c2008  x, 215 p. ; 22 cm
シリーズ名: Student mathematical library ; v. 44
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Vector spaces
Linear operators and matrices
Duality of vector spaces
Determinants Invariant subspaces
Operators on inner-product spaces
Structure theorems
Additional topics
Appendix
Index
Vector spaces
Linear operators and matrices
Duality of vector spaces
6.

図書

図書
Carl Meyer
出版情報: Philadelphia : Society for Industrial and Applied Mathematics (SIAM), c2000  2 v. ; 25 cm.
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Preface
Linear Equations / 1.:
Introduction / 1.1:
Gaussian Elimination and Matrices / 1.2:
Gauss-Jordan Method / 1.3:
Two-Point Boundary Value Problems / 1.4:
Making Gaussian Elimination Work / 1.5:
Ill-Conditioned Systems / 1.6:
Rectangular Systems and Echelon Forms / 2.:
Row Echelon Form and Rank / 2.1:
Reduced Row Echelon Form / 2.2:
Consistency of Linear Systems / 2.3:
Homogeneous Systems / 2.4:
Nonhomogeneous Systems / 2.5:
Electrical Circuits / 2.6:
Matrix Algebra / 3.:
From Ancient China to Arthur Cayley / 3.1:
Addition and Transposition / 3.2:
Linearity / 3.3:
Why Do It This Way / 3.4:
Matrix Multiplication / 3.5:
Properties of Matrix Multiplication / 3.6:
Matrix Inversion / 3.7:
Inverses of Sums and Sensitivity / 3.8:
Elementary Matrices and Equivalence / 3.9:
The LU Factorization / 3.10:
Vector Spaces / 4.:
Spaces and Subspaces / 4.1:
Four Fundamental Subspaces / 4.2:
Linear Independence / 4.3:
Basis and Dimension / 4.4:
More about Rank / 4.5:
Classical Least Squares / 4.6:
Linear Transformations / 4.7:
Change of Basis and Similarity / 4.8:
Invariant Subspaces / 4.9:
Norms, Inner Products, and Orthogonality / 5.:
Vector Norms / 5.1:
Matrix Norms / 5.2:
Inner-Product Spaces / 5.3:
Orthogonal Vectors / 5.4:
Gram-Schmidt Procedure / 5.5:
Unitary and Orthogonal Matrices / 5.6:
Orthogonal Reduction / 5.7:
Discrete Fourier Transform / 5.8:
Complementary Subspaces / 5.9:
Range-Nullspace Decomposition / 5.10:
Orthogonal Decomposition / 5.11:
Singular Value Decomposition / 5.12:
Orthogonal Projection / 5.13:
Why Least Squares? / 5.14:
Angles between Subspaces / 5.15:
Determinants / 6.:
Additional Properties of Determinants / 6.1:
Eigenvalues and Eigenvectors / 7.:
Elementary Properties of Eigensystems / 7.1:
Diagonalization by Similarity Transformations / 7.2:
Functions of Diagonalizable Matrices / 7.3:
Systems of Differential Equations / 7.4:
Normal Matrices / 7.5:
Positive Definite Matrices / 7.6:
Nilpotent Matrices and Jordan Structure / 7.7:
Jordan Form / 7.8:
Functions of Nondiagonalizable Matrices / 7.9:
Difference Equations, Limits, and Summability / 7.10:
Minimum Polynomials and Krylov Methods / 7.11:
Perron-Frobenius Theory / 8.:
Positive Matrices / 8.1:
Nonnegative Matrices / 8.3:
Stochastic Matrices and Markov Chains / 8.4:
Index
Preface
Linear Equations / 1.:
Introduction / 1.1:
7.

図書

図書
Morris W. Hirsch, Stephen Smale, Robert L. Devaney
出版情報: San Diego ; Tokyo : Elsevier Academic Press, c2004  xiv, 417 p. ; 24 cm
シリーズ名: Pure and applied mathematics ; v. 60
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Preface
First-Order Equations / Chapter 1:
The Simplest Example / 1.1:
The Logistic Population Model / 1.2:
Constant Harvesting and Bifurcations / 1.3:
Periodic Harvesting and Periodic Solutions / 1.4:
Computing the Poincare Map / 1.5:
Exploration: A Two-Parameter Family / 1.6:
Planar Linear Systems / Chapter 2:
Second-Order Differential Equations / 2.1:
Planar Systems / 2.2:
Preliminaries from Algebra / 2.3:
Eigenvalues and Eigenvectors / 2.4:
Solving Linear Systems / 2.6:
The Linearity Principle / 2.7:
Phase Portraits for Planar Systems / Chapter 3:
Real Distinct Eigenvalues / 3.1:
Complex Eigenvalues / 3.2:
Repeated Eigenvalues / 3.3:
Changing Coordinates / 3.4:
Classification of Planar Systems / Chapter 4:
The Trace-Determinant Plane / 4.1:
Dynamical Classification / 4.2:
Exploration: A 3D Parameter Space / 4.3:
Higher Dimensional Linear Algebra / Chapter 5:
Preliminaries from Linear Algebra / 5.1:
Bases and Subspaces / 5.2:
Genericity / 5.5:
Higher Dimensional Linear Systems / Chapter 6:
Distinct Eigenvalues / 6.1:
Harmonic Oscillators / 6.2:
The Exponential of a Matrix / 6.3:
Nonautonomous Linear Systems / 6.5:
Nonlinear Systems / Chapter 7:
Dynamical Systems / 7.1:
The Existence and Uniqueness Theorem / 7.2:
Continuous Dependence of Solutions / 7.3:
The Variational Equation / 7.4:
Exploration: Numerical Methods / 7.5:
Equilibria in Nonlinear Systems / Chapter 8:
Some Illustrative Examples / 8.1:
Nonlinear Sinks and Sources / 8.2:
Saddles / 8.3:
Stability / 8.4:
Bifurcations / 8.5:
Exploration: Complex Vector Fields / 8.6:
Global Nonlinear Techniques / Chapter 9:
Nullclines / 9.1:
Stability of Equilibria / 9.2:
Gradient Systems / 9.3:
Hamiltonian Systems / 9.4:
Exploration: The Pendulum with Constant Forcing / 9.5:
Closed Orbits and Limit Sets / Chapter 10:
Limit Sets / 10.1:
Local Sections and Flow Boxes / 10.2:
The Poincare Map / 10.3:
Monotone Sequences in Planar Dynamical Systems / 10.4:
The Poincare-Bendixson Theorem / 10.5:
Applications of Poincare-Bendixson / 10.6:
Exploration: Chemical Reactions That Oscillate / 10.7:
Applications in Biology / Chapter 11:
Infectious Diseases / 11.1:
Predator/Prey Systems / 11.2:
Competitive Species / 11.3:
Exploration: Competition and Harvesting / 11.4:
Applications in Circuit Theory / Chapter 12:
An RLC Circuit / 12.1:
The Lienard Equation / 12.2:
The van der Pol Equation / 12.3:
A Hopf Bifurcation / 12.4:
Exploration: Neurodynamics / 12.5:
Applications in Mechanics / Chapter 13:
Newton's Second Law / 13.1:
Conservative Systems / 13.2:
Central Force Fields / 13.3:
The Newtonian Central Force System / 13.4:
Kepler's First Law / 13.5:
The Two-Body Problem / 13.6:
Blowing Up the Singularity / 13.7:
Exploration: Other Central Force Problems / 13.8:
Exploration: Classical Limits of Quantum Mechanical Systems / 13.9:
The Lorenz System / Chapter 14:
Introduction to the Lorenz System / 14.1:
Elementary Properties of the Lorenz System / 14.2:
The Lorenz Attractor / 14.3:
A Model for the Lorenz Attractor / 14.4:
The Chaotic Attractor / 14.5:
Exploration: The Rossler Attractor / 14.6:
Discrete Dynamical Systems / Chapter 15:
Introduction to Discrete Dynamical Systems / 15.1:
The Discrete Logistic Model / 15.2:
Chaos / 15.4:
Symbolic Dynamics / 15.5:
The Shift Map / 15.6:
The Cantor Middle-Thirds Set / 15.7:
Exploration: Cubic Chaos / 15.8:
Exploration: The Orbit Diagram / 15.9:
Homoclinic Phenomena / Chapter 16:
The Shil'nikov System / 16.1:
The Horseshoe Map / 16.2:
The Double Scroll Attractor / 16.3:
Homoclinic Bifurcations / 16.4:
Exploration: The Chua Circuit / 16.5:
Existence and Uniqueness Revisited / Chapter 17:
Proof of Existence and Uniqueness / 17.1:
Continuous Dependence on Initial Conditions / 17.3:
Extending Solutions / 17.4:
Nonautonomous Systems / 17.5:
Differentiability of the Flow / 17.6:
Bibliography
Index
Preface
First-Order Equations / Chapter 1:
The Simplest Example / 1.1:
8.

図書

図書
Peter Deuflhard
出版情報: Berlin ; Tokyo : Springer, c2004  xii, 424 p. ; 24 cm
シリーズ名: Springer series in computational mathematics ; 35
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Outline of Contents
Introduction / 1:
Newton-Raphson Method for Scalar Equations / 1.1:
Newton's Method for General Nonlinear Problems / 1.2:
Classical convergence theorems revisited / 1.2.1:
Affine invariance and Lipschitz conditions / 1.2.2:
The algorithmic paradigm / 1.2.3:
A Roadmap of Newton-type Methods / 1.3:
Adaptive Inner Solvers for Inexact Newton Methods / 1.4:
Residual norm minimization: GMRES / 1.4.1:
Energy norm minimization: PCG / 1.4.2:
Error norm minimization: CGNE / 1.4.3:
Error norm reduction: GBIT / 1.4.4:
Linear multigrid methods / 1.4.5:
Exercises
Algebraic Equations / Part I:
Systems of Equations: Local Newton Methods / 2:
Error Oriented Algorithms / 2.1:
Ordinary Newton method / 2.1.1:
Simplified Newton method / 2.1.2:
Newton-like methods / 2.1.3:
Broyden's 'good' rank-1 updates / 2.1.4:
Inexact Newton-ERR methods / 2.1.5:
Residual Based Algorithms / 2.2:
Broyden's 'bad' rank-1 updates / 2.2.1:
Inexact Newton-RES method / 2.2.4:
Convex Optimization / 2.3:
Inexact Newton-PCG method / 2.3.1:
Systems of Equations: Global Newton Methods / 3:
Globalization Concepts / 3.1:
Componentwise convex mappings / 3.1.1:
Steepest descent methods / 3.1.2:
Trust region concepts / 3.1.3:
Newton path / 3.1.4:
Residual Based Descent / 3.2:
Affine contravariant convergence analysis / 3.2.1:
Adaptive trust region strategies / 3.2.2:
Error Oriented Descent / 3.2.3:
General level functions / 3.3.1:
Natural level function / 3.3.2:
Convex Functional Descent / 3.3.3:
Affine conjugate convergence analysis / 3.4.1:
Least Squares Problems: Gauss-Newton Methods / 3.4.2:
Linear Least Squares Problems / 4.1:
Unconstrained problems / 4.1.1:
Equality constrained problems / 4.1.2:
Local Gauss-Newton methods / 4.2:
Global Gauss-Newton methods / 4.2.2:
Adaptive trust region strategy / 4.2.3:
Local convergence results / 4.3:
Local Gauss-Newton algorithms / 4.3.2:
Global convergence results / 4.3.3:
Adaptive rank strategies / 4.3.4:
Underdetermined Systems of Equations / 4.4:
Local quasi-Gauss-Newton method / 4.4.1:
Global Gauss-Newton method / 4.4.2:
Parameter Dependent Systems: Continuation Methods / 5:
Newton Continuation Methods / 5.1:
Classification of continuation methods / 5.1.1:
Affine covariant feasible stepsizes / 5.1.2:
Adaptive pathfollowing algorithms / 5.1.3:
Gauss-Newton Continuation Method / 5.2:
Discrete tangent continuation beyond turning points / 5.2.1:
Adaptive stepsize control / 5.2.2:
Computation of Simple Bifurcations / 5.3:
Augmented systems for critical points / 5.3.1:
Newton-like algorithm for simple bifurcations / 5.3.2:
Branching-off algorithm / 5.3.3:
Differential Equations / Part II:
Stiff ODE Initial Value Problems / 6:
Affine Similar Linear Contractivity / 6.1:
Nonstiff versus Stiff Initial Value Problems / 6.2:
Picard iteration versus Newton iteration / 6.2.1:
Newton-type uniqueness theorems / 6.2.2:
Uniqueness Theorems for Implicit One-step Methods / 6.3:
Pseudo-transient Continuation for Steady State Problems / 6.4:
Exact pseudo-transient continuation / 6.4.1:
Inexact pseudo-transient continuation / 6.4.2:
ODE Boundary Value Problems / 7:
Multiple Shooting for Timelike BVPs / 7.1:
Cyclic linear systems / 7.1.1:
Realization of Newton methods / 7.1.2:
Realization of continuation methods / 7.1.3:
Parameter Identification in ODEs / 7.2:
Periodic Orbit Computation / 7.3:
Single orbit computation / 7.3.1:
Orbit continuation methods / 7.3.2:
Fourier collocation method / 7.3.3:
Polynomial Collocation for Spacelike BVPs / 7.4:
Discrete versus continuous solutions / 7.4.1:
Quasilinearization as inexact Newton method / 7.4.2:
PDE Boundary Value Problems / 8:
Asymptotic Mesh Independence / 8.1:
Global Discrete Newton Methods / 8.2:
General PDEs / 8.2.1:
Elliptic PDEs / 8.2.2:
Inexact Newton Multilevel FEM for Elliptic PDEs / 8.3:
Local Newton-Galerkin methods / 8.3.1:
Global Newton-Galerkin methods / 8.3.2:
References
Software
Index
Outline of Contents
Introduction / 1:
Newton-Raphson Method for Scalar Equations / 1.1:
9.

図書

図書
Steven Roman
出版情報: New York : Springer, c2005  xvi, 482 p. ; 24 cm
シリーズ名: Graduate texts in mathematics ; 135
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Preface to the Second Edition
Preface to the First Edition
Preliminaries
Algebraic Structures / Part 1:
Basic Linear Algebra / Part I:
Vector Spaces / 1:
Subspaces
Direct Sums
Spanning Sets and Linear Independence
The Dimension of a Vector Space
Ordered Bases and Coordinate Matrices
The Row and Column Spaces of a Matrix
The Complexification of a Real Vector Space
Exercises
Linear Transformations / 2:
Isomorphisms
The Kernel and Image of a Linear Transformation
Linear Transformations from F[superscript n] to F[superscript m]
The Rank Plus Nullity Theorem
Change of Basis Matrices
The Matrix of a Linear Transformation
Change of Bases for Linear Transformations
Equivalence of Matrices
Similarity of Matrices
Similarity of Operators
Invariant Subspaces and Reducing Paris
Topological Vector Spaces
Linear Operators on V[superscript C]
The Isomorphism Theorems / 3:
Quotient Spaces
The Universal Property of Quotients and the First Isomorphism Theorem
Quotient Spaces, Complements and Codimension
Additional Isomorphism Theorems
Linear Functionals
Dual Bases
Reflexivity
Annihilators
Operator Adjoints
Modules I: Basic Properties / 4:
Modules
Motivation
Submodules
Spanning Sets
Linear Independence
Torsion Elements
Free Modules
Homomorphisms
Quotient Modules
The Correspondence and Isomorphism Theorems
Direct Sums and Direct Summands
Modules Are Not As Nice As Vector Spaces
Modules II: Free and Noetherian Modules / 5:
The Rank of a Free Module
Free Modules and Epimorphisms
Noetherian Modules
The Hilbert Basis Theorem
Modules over a Principal Ideal Domain / 6:
Annihilators and Orders
Cyclic Modules
Free Modules over a Principal Ideal Domain
Torsion-Free and Free Modules
Prelude to Decomposition: Cyclic Modules
The First Decomposition
A Look Ahead
The Primary Decomposition
The Cyclic Decomposition of a Primary Module
The Primary Cyclic Decomposition Theorem
The Invariant Factor Decomposition
The Structure of a Linear Operator / 7:
A Brief Review
The Module Associated with a Linear Operator
Orders and the Minimal Polynomial
Cyclic Submodules and Cyclic Subspaces
Summary
The Decomposition of V[subscript tau]
The Rational Canonical Form
Eigenvalues and Eigenvectors / 8:
The Characteristic Polynomial of an Operator
Geometric and Algebraic Multiplicities
The Jordan Canonical Form
Triangularizability and Schur's Lemma
Diagonalizable Operators
Projections
The Algebra of Projections
Resolutions of the Identity
Spectral Resolutions
Projections and Invariance
Real and Complex Inner Product Spaces / 9:
Norm and Distance
Isometries
Orthogonality
Orthogonal and Orthonormal Sets
The Projection Theorem and Best Approximations
Orthogonal Direct Sums
The Riesz Representation Theorem
Structure Theory for Normal Operators / 10:
The Adjoint of a Linear Operator
Unitary Diagonalizability
Normal Operators
Special Types of Normal Operators
Self-Adjoint Operators
Unitary Operators and Isometries
The Structure of Normal Operators
Matrix Versions
Orthogonal Projections
Orthogonal Resolutions of the Identity
The Spectral Theorem
Spectral Resolutions and Functional Calculus
Positive Operators
The Polar Decomposition of an Operator
Topics / Part II:
Metric Vector Spaces: The Theory of Bilinear Forms / 11:
Symmetric, Skew-Symmetric and Alternate Forms
The Matrix of a Bilinear Form
Quadratic Forms
Orthogonal Complements and Orthogonal Direct Sums
Hyperbolic Spaces
Nonsingular Completions of a Subspace
The Witt Theorems: A Preview
The Classification Problem for Metric Vector Spaces
Symplectic Geometry
The Structure of Orthogonal Geometries: Orthogonal Bases
The Classification of Orthogonal Geometries: Canonical Forms
The Orthogonal Group
The Witt's Theorems for Orthogonal Geometries
Maximal Hyperbolic Subspaces of an Orthogonal Geometry
Metric Spaces / 12:
The Definition
Open and Closed Sets
Convergence in a Metric Space
The Closure of a Set
Dense Subsets
Continuity
Completeness
The Completion of a Metric Space
Hilbert Spaces / 13:
Infinite Series
An Approximation Problem
Hilbert Bases
Fourier Expansions
A Characterization of Hilbert Bases
Hilbert Dimension
A Characterization of Hilbert Spaces
Tensor Products / 14:
Universality
Bilinear Maps
When Is a Tensor Product Zero?
Coordinate Matrices and Rank
Characterizing Vectors in a Tensor Product
Defining Linear Transformations on a Tensor Product
The Tensor Product of Linear Transformations
Change of Base Field
Multilinear Maps and Iterated Tensor Products
Tensor Spaces
Special Multilinear Maps
Graded Algebras
The Symmetric Tensor Algebra
The Antisymmetric Tensor Algebra: The Exterior Product Space
The Determinant
Positive Solutions to Linear Systems: Convexity and Separation / 15:
Convex, Closed and Compact Sets
Convex Hulls
Linear and Affine Hyperplanes
Separation
Affine Geometry / 16:
Affine Combinations
Affine Hulls
The Lattice of Flats
Affine Independence
Affine Transformations
Projective Geometry
Operator Factorizations: QR and Singular Value / 17:
The QR Decomposition
Singular Values
The Moore-Penrose Generalized Inverse
Least Squares Approximation
The Umbral Calculus / 18:
Formal Power Series
The Umbral Algebra
Formal Power Series as Linear Operators
Sheffer Sequences
Examples of Sheffer Sequences
Umbral Operators and Umbral Shifts
Continuous Operators on the Umbral Algebra
Umbral Operators and Automorphisms of the Umbral Algebra
Umbral Shifts and Derivations of the Umbral Algebra
The Transfer Formulas
A Final Remark
References
Index
Preface to the Second Edition
Preface to the First Edition
Preliminaries
10.

図書

図書
Michael C. Ferris, Olvi L. Mangasarian, Stephen J. Wright
出版情報: Philadelphia : Society for Industrial and Applied Mathematics : Mathematical Programming Society, c2007  xi, 266 p. ; 26 cm
シリーズ名: MPS-SIAM series on optimization ; 7
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