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15. Linear and Multilinear Algebra; Matrix Theory

This volume develops vector spaces, linear maps, matrices, and multilinear structures. It serves as a core toolkit for nearly all areas of mathematics, physics, and computation. Part I. Vector Spaces Chapter 1. Vector Spaces 1.1 Definitions and examples 1.2 Subspaces 1.3 Linear combinations 1.4 Span and generation 1.5 Linear independence Chapter 2. Bases and Dimension 2.1 Bases 2.2 Existence of bases 2.3 Dimension 2.4 Coordinate representations 2.5 Change of basis...

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Appendices

Appendix Title A Set Theory and Logic B Proof Techniques C Real and Complex Numbers D Polynomial Algebra E Calculus Review F Numerical Computation G Mathematical Notation H Historical Notes I Glossary J Theorem Index K Symbol Index

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X. Specialized Topics

Chapter Title 127 Complex Vector Spaces 128 Finite Fields 129 Linear Algebra over Arbitrary Fields 130 Modules and Linear Algebra 131 Category-Theoretic Perspective 132 Convex Geometry 133 Random Matrices 134 Numerical Optimization 135 Operator Theory 136 Spectral Graph Theory 137 Compressed Sensing 138 Tensor Decompositions 139 Geometric Algebra 140 Modern Applications in AI

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VI. Matrix Decompositions

Chapter Title 75 LU Decomposition 76 PLU Decomposition 77 Cholesky Decomposition 78 QR Decomposition 79 Schur Decomposition 80 Singular Value Decomposition 81 Polar Decomposition 82 Hessenberg Form 83 Tridiagonalization 84 Canonical Matrix Forms

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II. Vector Spaces

Chapter Title 17 Vector Spaces 18 Subspaces 19 Span and Linear Combination 20 Linear Independence 21 Basis 22 Dimension 23 Coordinate Systems 24 Change of Basis 25 Row Space and Column Space 26 Null Space 27 Quotient Spaces 28 Dual Spaces 29 Annihilators 30 Direct Sums 31 Affine Spaces

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IV. Inner Product Spaces

Chapter Title 45 Inner Products 46 Norms and Metrics 47 Orthogonality 48 Orthogonal Complements 49 Orthonormal Bases 50 Gram-Schmidt Orthogonalization 51 Orthogonal Projections 52 Least Squares Problems 53 QR Factorization 54 Unitary and Orthogonal Matrices 55 Hermitian Spaces 56 Positive Definite Matrices 57 Quadratic Forms 58 Sylvester's Law of Inertia

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VII. Numerical Linear Algebra

Chapter Title 85 Floating Point Arithmetic 86 Conditioning and Stability 87 Error Analysis 88 Iterative Methods for Linear Systems 89 Jacobi Method 90 Gauss-Seidel Method 91 Conjugate Gradient Method 92 Krylov Subspaces 93 Power Iteration 94 QR Algorithm 95 Sparse Matrices 96 Structured Matrices 97 Randomized Linear Algebra

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Linear Algebra

Linear Algebra Part I. Foundations Chapter Title 1 What Linear Algebra Is 2 Scalars, Vectors, and Fields 3 Geometry of Euclidean Space 4 Linear Equations 5 Systems of Linear Equations 6 Matrices 7 Matrix Operations 8 Elementary Row Operations 9 Gaussian Elimination 10 Gauss-Jordan Elimination 11 Invertible Matrices 12 Rank and Nullity 13 Determinants 14 Cofactors and Adjugates 15 Block Matrices 16 Matrix Factorizations Overview Part II. Vector Spaces Chapter...

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IX. Applications

Chapter Title 109 Linear Regression 110 Optimization and Linear Algebra 111 Graphs and Adjacency Matrices 112 Markov Chains 113 Differential Equations 114 Fourier Series and Transforms 115 Signal Processing 116 Computer Graphics 117 Robotics and Kinematics 118 Control Theory 119 Quantum Mechanics 120 Machine Learning 121 Principal Component Analysis 122 PageRank and Network Analysis 123 Coding Theory 124 Cryptography 125 Finite Element Methods 126 Scientific Computing

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III. Linear Transformations

Chapter Title 32 Linear Transformations 33 Kernel and Image 34 Matrix Representation of Linear Maps 35 Composition of Transformations 36 Isomorphisms 37 Automorphisms 38 Projection Operators 39 Reflection Operators 40 Rotation Operators 41 Shears and Scalings 42 Similarity Transformations 43 Invariant Subspaces 44 Cyclic Subspaces

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V. Eigenvalues and Spectral Theory

Chapter Title 59 Eigenvalues 60 Eigenvectors 61 Characteristic Polynomial 62 Eigenspaces 63 Diagonalization 64 Spectral Theorem 65 Symmetric Matrices 66 Hermitian Operators 67 Normal Operators 68 Jordan Canonical Form 69 Minimal Polynomial 70 Cayley-Hamilton Theorem 71 Rational Canonical Form 72 Matrix Functions 73 Matrix Exponential 74 Perron-Frobenius Theory

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VIII. Advanced Structures

Chapter Title 98 Tensor Products 99 Exterior Algebra 100 Symmetric Algebra 101 Multilinear Maps 102 Bilinear Forms 103 Alternating Forms 104 Clifford Algebras 105 Lie Algebras 106 Representation Theory Basics 107 Infinite-Dimensional Vector Spaces 108 Functional Analysis Connections

Writes › Book › Linear Algebra ›

I. Foundations

Chapter Title 1 What Linear Algebra Is 2 Scalars, Vectors, and Fields 3 Geometry of Euclidean Space 4 Linear Equations 5 Systems of Linear Equations 6 Matrices 7 Matrix Operations 8 Elementary Row Operations 9 Gaussian Elimination 10 Gauss-Jordan Elimination 11 Invertible Matrices 12 Rank and Nullity 13 Determinants 14 Cofactors and Adjugates 15 Block Matrices 16 Matrix Factorizations Overview

#linear-algebra

15. Linear and Multilinear Algebra; Matrix Theory

Appendices

X. Specialized Topics

VI. Matrix Decompositions

II. Vector Spaces

IV. Inner Product Spaces

VII. Numerical Linear Algebra

Linear Algebra

IX. Applications

III. Linear Transformations

V. Eigenvalues and Spectral Theory

VIII. Advanced Structures

I. Foundations