Linear algebra and optimization with applications to machine learning (Record no. 35243)

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International Standard Book Number 9789811206399
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International Standard Book Number 9789811207716
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082 00 - DEWEY DECIMAL CLASSIFICATION NUMBER
Classification number 512.5
Edition number 23
Item number GAL
100 1# - MAIN ENTRY--PERSONAL NAME
Personal name Gallier, Jean H.,
245 10 - TITLE STATEMENT
Title Linear algebra and optimization with applications to machine learning
Statement of responsibility, etc Jean Gallier, Jocelyn Quaintance.
300 ## - PHYSICAL DESCRIPTION
Extent volume :
Other physical details illustrations (some color) ;
Dimensions 24 cm
505 1# - FORMATTED CONTENTS NOTE
Contents Volume I. Linear algebra for computer vision, robotics, and machine learning -- Volume II. Fundamentals of optimization theory with applications to machine learning
Title 1 Introduction 11<br/>2 Vector Spaces, Bases, Linear Maps 15<br/>2.1 Motivations: Linear Combinations, Linear Independence, Rank . . . . . . . 15<br/>2.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27<br/>2.3 Indexed Families; the Sum Notation P<br/>i2I ai . . . . . . . . . . . . . . . . . . 34<br/>2.4 Linear Independence, Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 40<br/>2.5 Bases of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46<br/>2.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53<br/>2.7 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58<br/>2.8 Linear Forms and the Dual Space . . . . . . . . . . . . . . . . . . . . . . . . 65<br/>2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68<br/>2.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70<br/>3 Matrices and Linear Maps 77<br/>3.1 Representation of Linear Maps by Matrices . . . . . . . . . . . . . . . . . . 77<br/>3.2 Composition of Linear Maps and Matrix Multiplication . . . . . . . . . . . 82<br/>3.3 Change of Basis Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87<br/>3.4 The E↵ect of a Change of Bases on Matrices . . . . . . . . . . . . . . . . . 90<br/>3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94<br/>3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94<br/>4 Haar Bases, Haar Wavelets, Hadamard Matrices 101<br/>4.1 Introduction to Signal Compression Using Haar Wavelets . . . . . . . . . . 101<br/>4.2 Haar Matrices, Scaling Properties of Haar Wavelets . . . . . . . . . . . . . . 103<br/>4.3 Kronecker Product Construction of Haar Matrices . . . . . . . . . . . . . . 108<br/>4.4 Multiresolution Signal Analysis with Haar Bases . . . . . . . . . . . . . . . 110<br/>4.5 Haar Transform for Digital Images . . . . . . . . . . . . . . . . . . . . . . . 112<br/>4.6 Hadamard Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119<br/>4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121<br/>4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121<br/>5 Direct Sums, Rank-Nullity Theorem, Ane Maps 125<br/>5.1 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125<br/>5.2 Sums and Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126<br/>5.3 The Rank-Nullity Theorem; Grassmann’s Relation . . . . . . . . . . . . . . 131<br/>5.4 Ane Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137<br/>5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144<br/>5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145<br/>6 Determinants 153<br/>6.1 Permutations, Signature of a Permutation . . . . . . . . . . . . . . . . . . . 153<br/>6.2 Alternating Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 158<br/>6.3 Definition of a Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 162<br/>6.4 Inverse Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . . 170<br/>6.5 Systems of Linear Equations and Determinants . . . . . . . . . . . . . . . . 173<br/>6.6 Determinant of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . 175<br/>6.7 The Cayley–Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 176<br/>6.8 Permanents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181<br/>6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183<br/>6.10 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185<br/>6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185<br/>7 Gaussian Elimination, LU, Cholesky, Echelon Form 191<br/>7.1 Motivating Example: Curve Interpolation . . . . . . . . . . . . . . . . . . . 191<br/>7.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195<br/>7.3 Elementary Matrices and Row Operations . . . . . . . . . . . . . . . . . . . 200<br/>7.4 LU-Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203<br/>7.5 P A = LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209<br/>7.6 Proof of Theorem 7.5 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217<br/>7.7 Dealing with Roundo↵ Errors; Pivoting Strategies . . . . . . . . . . . . . . . 223<br/>7.8 Gaussian Elimination of Tridiagonal Matrices . . . . . . . . . . . . . . . . . 224<br/>7.9 SPD Matrices and the Cholesky Decomposition . . . . . . . . . . . . . . . . 226<br/>7.10 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 235<br/>7.11 RREF, Free Variables, Homogeneous Systems . . . . . . . . . . . . . . . . . 241<br/>7.12 Uniqueness of RREF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244<br/>7.13 Solving Linear Systems Using RREF . . . . . . . . . . . . . . . . . . . . . . 246<br/>7.14 Elementary Matrices and Columns Operations . . . . . . . . . . . . . . . . 253<br/>7.15 Transvections and Dilatations ~ . . . . . . . . . . . . . . . . . . . . . . . . 254<br/>7.16 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259<br/>7.17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261<br/>8 Vector Norms and Matrix Norms 273<br/>8.1 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273<br/>8.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284<br/>8.3 Subordinate Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289<br/>8.4 Inequalities Involving Subordinate Norms . . . . . . . . . . . . . . . . . . . 296<br/>8.5 Condition Numbers of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 298<br/>8.6 An Application of Norms: Inconsistent Linear Systems . . . . . . . . . . . . 307<br/>8.7 Limits of Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . 308<br/>8.8 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311<br/>8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314<br/>8.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316<br/>9 Iterative Methods for Solving Linear Systems 323<br/>9.1 Convergence of Sequences of Vectors and Matrices . . . . . . . . . . . . . . 323<br/>9.2 Convergence of Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . 326<br/>9.3 Methods of Jacobi, Gauss–Seidel, and Relaxation . . . . . . . . . . . . . . . 328<br/>9.4 Convergence of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 336<br/>9.5 Convergence Methods for Tridiagonal Matrices . . . . . . . . . . . . . . . . 339<br/>9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343<br/>9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344<br/>10 The Dual Space and Duality 347<br/>10.1 The Dual Space E⇤ and Linear Forms . . . . . . . . . . . . . . . . . . . . . 347<br/>10.2 Pairing and Duality Between E and E⇤ . . . . . . . . . . . . . . . . . . . . 354<br/>10.3 The Duality Theorem and Some Consequences . . . . . . . . . . . . . . . . 359<br/>10.4 The Bidual and Canonical Pairings . . . . . . . . . . . . . . . . . . . . . . . 364<br/>10.5 Hyperplanes and Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . 366<br/>10.6 Transpose of a Linear Map and of a Matrix . . . . . . . . . . . . . . . . . . 367<br/>10.7 Properties of the Double Transpose . . . . . . . . . . . . . . . . . . . . . . . 372<br/>10.8 The Four Fundamental Subspaces . . . . . . . . . . . . . . . . . . . . . . . 374<br/>10.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377<br/>10.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378<br/>11 Euclidean Spaces 381<br/>11.1 Inner Products, Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . 381<br/>11.2 Orthogonality and Duality in Euclidean Spaces . . . . . . . . . . . . . . . . 390<br/>11.3 Adjoint of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397<br/>11.4 Existence and Construction of Orthonormal Bases . . . . . . . . . . . . . . 400<br/>11.5 Linear Isometries (Orthogonal Transformations) . . . . . . . . . . . . . . . . 407<br/>11.6 The Orthogonal Group, Orthogonal Matrices . . . . . . . . . . . . . . . . . 410<br/>11.7 The Rodrigues Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412<br/>11.8 QR-Decomposition for Invertible Matrices . . . . . . . . . . . . . . . . . . . 415<br/>11.9 Some Applications of Euclidean Geometry . . . . . . . . . . . . . . . . . . . 420<br/>11.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421<br/>11.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423<br/>12 QR-Decomposition for Arbitrary Matrices 435<br/>12.1 Orthogonal Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435<br/>12.2 QR-Decomposition Using Householder Matrices . . . . . . . . . . . . . . . . 440<br/>12.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450<br/>12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450<br/>13 Hermitian Spaces 457<br/>13.1 Hermitian Spaces, Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . 457<br/>13.2 Orthogonality, Duality, Adjoint of a Linear Map . . . . . . . . . . . . . . . 466<br/>13.3 Linear Isometries (Also Called Unitary Transformations) . . . . . . . . . . . 471<br/>13.4 The Unitary Group, Unitary Matrices . . . . . . . . . . . . . . . . . . . . . 473<br/>13.5 Hermitian Reflections and QR-Decomposition . . . . . . . . . . . . . . . . . 476<br/>13.6 Orthogonal Projections and Involutions . . . . . . . . . . . . . . . . . . . . 481<br/>13.7 Dual Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484<br/>13.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491<br/>13.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492<br/>14 Eigenvectors and Eigenvalues 497<br/>14.1 Eigenvectors and Eigenvalues of a Linear Map . . . . . . . . . . . . . . . . . 497<br/>14.2 Reduction to Upper Triangular Form . . . . . . . . . . . . . . . . . . . . . . 505<br/>14.3 Location of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509<br/>14.4 Conditioning of Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . 512<br/>14.5 Eigenvalues of the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . 515<br/>14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517<br/>14.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518<br/>15 Unit Quaternions and Rotations in SO(3) 529<br/>15.1 The Group SU(2) and the Skew Field H of Quaternions . . . . . . . . . . . 529<br/>15.2 Representation of Rotation in SO(3) By Quaternions in SU(2) . . . . . . . 531<br/>15.3 Matrix Representation of the Rotation rq . . . . . . . . . . . . . . . . . . . 536<br/>15.4 An Algorithm to Find a Quaternion Representing a Rotation . . . . . . . . 538<br/>15.5 The Exponential Map exp: su(2) ! SU(2) . . . . . . . . . . . . . . . . . . 541<br/>15.6 Quaternion Interpolation ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . 543<br/>15.7 Nonexistence of a “Nice” Section from SO(3) to SU(2) . . . . . . . . . . . . 545<br/>15.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547<br/>15.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548<br/>16 Spectral Theorems 551<br/>16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551<br/>16.2 Normal Linear Maps: Eigenvalues and Eigenvectors . . . . . . . . . . . . . . 551<br/>16.3 Spectral Theorem for Normal Linear Maps . . . . . . . . . . . . . . . . . . . 557<br/>16.4 Self-Adjoint and Other Special Linear Maps . . . . . . . . . . . . . . . . . . 562<br/>16.5 Normal and Other Special Matrices . . . . . . . . . . . . . . . . . . . . . . . 568<br/>16.6 Rayleigh–Ritz Theorems and Eigenvalue Interlacing . . . . . . . . . . . . . 571<br/>16.7 The Courant–Fischer Theorem; Perturbation Results . . . . . . . . . . . . . 576<br/>16.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579<br/>16.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580<br/>17 Computing Eigenvalues and Eigenvectors 587<br/>17.1 The Basic QR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589<br/>17.2 Hessenberg Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595<br/>17.3 Making the QR Method More Ecient Using Shifts . . . . . . . . . . . . . 601<br/>17.4 Krylov Subspaces; Arnoldi Iteration . . . . . . . . . . . . . . . . . . . . . . 606<br/>17.5 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610<br/>17.6 The Hermitian Case; Lanczos Iteration . . . . . . . . . . . . . . . . . . . . . 611<br/>17.7 Power Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612<br/>17.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614<br/>17.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615<br/>18 Graphs and Graph Laplacians; Basic Facts 617<br/>18.1 Directed Graphs, Undirected Graphs, Weighted Graphs . . . . . . . . . . . 620<br/>18.2 Laplacian Matrices of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 627<br/>18.3 Normalized Laplacian Matrices of Graphs . . . . . . . . . . . . . . . . . . . 631<br/>18.4 Graph Clustering Using Normalized Cuts . . . . . . . . . . . . . . . . . . . 635<br/>18.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637<br/>18.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638<br/>19 Spectral Graph Drawing 641<br/>19.1 Graph Drawing and Energy Minimization . . . . . . . . . . . . . . . . . . . 641<br/>19.2 Examples of Graph Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . 644<br/>19.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648<br/>20 Singular Value Decomposition and Polar Form 651<br/>20.1 Properties of f ⇤ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651<br/>20.2 Singular Value Decomposition for Square Matrices . . . . . . . . . . . . . . 655<br/>20.3 Polar Form for Square Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 658<br/>20.4 Singular Value Decomposition for Rectangular Matrices . . . . . . . . . . . 661<br/>20.5 Ky Fan Norms and Schatten Norms . . . . . . . . . . . . . . . . . . . . . . 664<br/>20.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665<br/>20.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665<br/>21 Applications of SVD and Pseudo-Inverses 669<br/>21.1 Least Squares Problems and the Pseudo-Inverse . . . . . . . . . . . . . . . . 669<br/>21.2 Properties of the Pseudo-Inverse . . . . . . . . . . . . . . . . . . . . . . . . 676<br/>21.3 Data Compression and SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . 681<br/>21.4 Principal Components Analysis (PCA) . . . . . . . . . . . . . . . . . . . . . 683<br/>21.5 Best Ane Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 694<br/>21.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698<br/>21.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699<br/>22 Annihilating Polynomials; Primary Decomposition 703<br/>22.1 Basic Properties of Polynomials; Ideals, GCD’s . . . . . . . . . . . . . . . . 705<br/>22.2 Annihilating Polynomials and the Minimal Polynomial . . . . . . . . . . . . 710<br/>22.3 Minimal Polynomials of Diagonalizable Linear Maps . . . . . . . . . . . . . 711<br/>22.4 Commuting Families of Linear Maps . . . . . . . . . . . . . . . . . . . . . . 714<br/>22.5 The Primary Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . 717<br/>22.6 Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724<br/>22.7 Nilpotent Linear Maps and Jordan Form . . . . . . . . . . . . . . . . . . . . 726<br/>22.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732<br/>22.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733<br/>Bibliography 735
520 ## - SUMMARY, ETC.
Summary, etc "This book provides the mathematical fundamentals of linear algebra to practicers in computer vision, machine learning, robotics, applied mathematics, and electrical engineering. By only assuming a knowledge of calculus, the authors develop, in a rigorous yet down to earth manner, the mathematical theory behind concepts such as: vectors spaces, bases, linear maps, duality, Hermitian spaces, the spectral theorems, SVD, and the primary decomposition theorem. At all times, pertinent real-world applications are provided. This book includes the mathematical explanations for the tools used which we believe that is adequate for computer scientists, engineers and mathematicians who really want to do serious research and make significant contributions in their respective fields"--
650 #0 - SUBJECT ADDED ENTRY--TOPICAL TERM
Topical term or geographic name as entry element Algebras, Linear.
650 #0 - SUBJECT ADDED ENTRY--TOPICAL TERM
Topical term or geographic name as entry element Machine learning
700 1# - ADDED ENTRY--PERSONAL NAME
Personal name Quaintance, Jocelyn,
942 ## - ADDED ENTRY ELEMENTS (KOHA)
Source of classification or shelving scheme Dewey Decimal Classification
Koha item type General Books
100 1# - MAIN ENTRY--PERSONAL NAME
Relator term author.
504 ## - BIBLIOGRAPHY, ETC. NOTE
Bibliography, etc Includes bibliographical references and index.
650 #0 - SUBJECT ADDED ENTRY--TOPICAL TERM
General subdivision Mathematics.
700 1# - ADDED ENTRY--PERSONAL NAME
Relator term author.
906 ## - LOCAL DATA ELEMENT F, LDF (RLIN)
a 7
b cbc
c orignew
d 1
e ecip
f 20
g y-gencatlg
Holdings
Withdrawn status Lost status Source of classification or shelving scheme Damaged status Not for loan Collection code Home library Location Shelving location Date of Cataloging Total Checkouts Full call number Barcode Date last seen Date checked out Price effective from Koha item type
    Dewey Decimal Classification     Non-fiction CUTN Central Library CUTN Central Library Sciences 06/07/2021 1 512.5 GAL 44012 15/07/2022 14/06/2022 06/07/2021 General Books