linear algebra and optimization with Applications to Machine Learning 1st edition by Jean Gallier, Jocelyn Quaintance ISBN

linear algebra and optimization with Applications to Machine Learning 1st edition by Jean Gallier, Jocelyn Quaintance  – Ebook PDF Instant Download/Delivery: 9811207712‎ , 978-9811207716
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Product details:

ISBN 10: 9811207712
ISBN 13:‎ 978-9811207716
Author: Jean Gallier, Jocelyn Quaintance 

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.

linear algebra and optimization with Applications to Machine Learning 1st Table of contents:

1. Introduction

2. Vector Spaces, Bases, Linear Maps

2.1 Motivations: Linear Combinations, Linear Independence and Rank

2.2 Vector Spaces

2.3 Indexed Families; the Sum Notation ∑i∈I ai

2.4 Linear Independence, Subspaces

2.5 Bases of a Vector Space

2.6 Matrices

2.7 Linear Maps

2.8 Linear Forms and the Dual Space

2.9 Summary

2.10 Problems

3. Matrices and Linear Maps

3.1 Representation of Linear Maps by Matrices

3.2 Composition of Linear Maps and Matrix Multiplication

3.3 Change of Basis Matrix

3.4 The Effect of a Change of Bases on Matrices

3.5 Summary

3.6 Problems

4. Haar Bases, Haar Wavelets, Hadamard Matrices

4.1 Introduction to Signal Compression Using Haar Wavelets

4.2 Haar Bases and Haar Matrices, Scaling Properties of Haar Wavelets

4.3 Kronecker Product Construction of Haar Matrices

4.4 Multiresolution Signal Analysis with Haar Bases

4.5 Haar Transform for Digital Images

4.6 Hadamard Matrices

4.7 Summary

4.8 Problems

5. Direct Sums, Rank-Nullity Theorem, Affine Maps

5.1 Direct Products

5.2 Sums and Direct Sums

5.3 The Rank-Nullity Theorem; Grassmann’s Relation

5.4 Affine Maps

5.5 Summary

5.6 Problems

6. Determinants

6.1 Permutations, Signature of a Permutation

6.2 Alternating Multilinear Maps

6.3 Definition of a Determinant

6.4 Inverse Matrices and Determinants

6.5 Systems of Linear Equations and Determinants

6.6 Determinant of a Linear Map

6.7 The Cayley–Hamilton Theorem

6.8 Permanents

6.9 Summary

6.10 Further Readings

6.11 Problems

7. Gaussian Elimination, LU-Factorization, Cholesky Factorization, Reduced Row Echelon Form

7.1 Motivating Example: Curve Interpolation

7.2 Gaussian Elimination

7.3 Elementary Matrices and Row Operations

7.4 LU-Factorization

7.5 PA = LU Factorization

7.6 Proof of Theorem 7.2 ⊛

7.7 Dealing with Roundoff Errors; Pivoting Strategies

7.8 Gaussian Elimination of Tridiagonal Matrices

7.9 SPD Matrices and the Cholesky Decomposition

7.10 Reduced Row Echelon Form (RREF)

7.11 RREF, Free Variables, and Homogenous Linear Systems

7.12 Uniqueness of RREF Form

7.13 Solving Linear Systems Using RREF

7.14 Elementary Matrices and Columns Operations

7.15 Transvections and Dilatations ⊛

7.16 Summary

7.17 Problems

8. Vector Norms and Matrix Norms

8.1 Normed Vector Spaces

8.2 Matrix Norms

8.3 Subordinate Norms

8.4 Inequalities Involving Subordinate Norms

8.5 Condition Numbers of Matrices

8.6 An Application of Norms: Solving Inconsistent Linear Systems

8.7 Limits of Sequences and Series

8.8 The Matrix Exponential

8.9 Summary

8.10 Problems

9. Iterative Methods for Solving Linear Systems

9.1 Convergence of Sequences of Vectors and Matrices

9.2 Convergence of Iterative Methods

9.3 Description of the Methods of Jacobi, Gauss–Seidel, and Relaxation

9.4 Convergence of the Methods of Gauss–Seidel and Relaxation

9.5 Convergence of the Methods of Jacobi, Gauss–Seidel, and Relaxation for Tridiagonal Matrices

9.6 Summary

9.7 Problems

10. The Dual Space and Duality

10.1 The Dual Space E* and Linear Forms

10.2 Pairing and Duality Between E and E*

10.3 The Duality Theorem and Some Consequences

10.4 The Bidual and Canonical Pairings

10.5 Hyperplanes and Linear Forms

10.6 Transpose of a Linear Map and of a Matrix

10.7 Properties of the Double Transpose

10.8 The Four Fundamental Subspaces

10.9 Summary

10.10 Problems

11. Euclidean Spaces

11.1 Inner Products, Euclidean Spaces

11.2 Orthogonality and Duality in Euclidean Spaces

11.3 Adjoint of a Linear Map

11.4 Existence and Construction of Orthonormal Bases

11.5 Linear Isometries (Orthogonal Transformations)

11.6 The Orthogonal Group, Orthogonal Matrices

11.7 The Rodrigues Formula

11.8 QR-Decomposition for Invertible Matrices

11.9 Some Applications of Euclidean Geometry

11.10 Summary

11.11 Problems

12. QR-Decomposition for Arbitrary Matrices

12.1 Orthogonal Reflections

12.2 QR-Decomposition Using Householder Matrices

12.3 Summary

12.4 Problems

13. Hermitian Spaces

13.1 Sesquilinear and Hermitian Forms, Pre-Hilbert Spaces and Hermitian Spaces

13.2 Orthogonality, Duality, Adjoint of a Linear Map

13.3 Linear Isometries (Also Called Unitary Transformations)

13.4 The Unitary Group, Unitary Matrices

13.5 Hermitian Reflections and QR-Decomposition

13.6 Orthogonal Projections and Involutions

13.7 Dual Norms

13.8 Summary

13.9 Problems

14. Eigenvectors and Eigenvalues

14.1 Eigenvectors and Eigenvalues of a Linear Map

14.2 Reduction to Upper Triangular Form

14.3 Location of Eigenvalues

14.4 Conditioning of Eigenvalue Problems

14.5 Eigenvalues of the Matrix Exponential

14.6 Summary

14.7 Problems

15. Unit Quaternions and Rotations in SO(3)

15.1 The Group SU(2) of Unit Quaternions and the Skew Field H of Quaternions

15.2 Representation of Rotations in SO(3) by Quaternions in SU(2)

15.3 Matrix Representation of the Rotation rq

15.4 An Algorithm to Find a Quaternion Representing a Rotation

15.5 The Exponential Map exp: su(2) → SU(2)

15.6 Quaternion Interpolation ⊛

15.7 Nonexistence of a “Nice” Section from SO(3) to SU(2)

15.8 Summary

15.9 Problems

16. Spectral Theorems in Euclidean and Hermitian Spaces

16.1 Introduction

16.2 Normal Linear Maps: Eigenvalues and Eigenvectors

16.3 Spectral Theorem for Normal Linear Maps

16.4 Self-Adjoint, Skew-Self-Adjoint, and Orthogonal Linear Maps

16.5 Normal and Other Special Matrices

16.6 Rayleigh–Ritz Theorems and Eigenvalue Interlacing

16.7 The Courant−Fischer Theorem; Perturbation Results

16.8 Summary

16.9 Problems

17. Computing Eigenvalues and Eigenvectors

17.1 The Basic QR Algorithm

17.2 Hessenberg Matrices

17.3 Making the QR Method More Efficient Using Shifts

17.4 Krylov Subspaces; Arnoldi Iteration

17.5 GMRES

17.6 The Hermitian Case; Lanczos Iteration

17.7 Power Methods

17.8 Summary

17.9 Problems

18. Graphs and Graph Laplacians; Basic Facts

18.1 Directed Graphs, Undirected Graphs, Incidence Matrices, Adjacency Matrices, Weighted Graphs

18.2 Laplacian Matrices of Graphs

18.3 Normalized Laplacian Matrices of Graphs

18.4 Graph Clustering Using Normalized Cuts

18.5 Summary

18.6 Problems

19. Spectral Graph Drawing

19.1 Graph Drawing and Energy Minimization

19.2 Examples of Graph Drawings

19.3 Summary

20. Singular Value Decomposition and Polar Form

20.1 Properties of f* ○ f

20.2 Singular Value Decomposition for Square Matrices

20.3 Polar Form for Square Matrices

20.4 Singular Value Decomposition for Rectangular Matrices

20.5 Ky Fan Norms and Schatten Norms

20.6 Summary

20.7 Problems

21. Applications of SVD and Pseudo-Inverses

21.1 Least Squares Problems and the Pseudo-Inverse

21.2 Properties of the Pseudo-Inverse

21.3 Data Compression and SVD

21.4 Principal Components Analysis (PCA)

21.5 Best Affine Approximation

21.6 Summary

21.7 Problems

22. Annihilating Polynomials and the Primary Decomposition

22.1 Basic Properties of Polynomials; Ideals, GCD’s

22.2 Annihilating Polynomials and the Minimal Polynomial

22.3 Minimal Polynomials of Diagonalizable Linear Maps

22.4 Commuting Families of Diagonalizable and Triangulable Maps

22.5 The Primary Decomposition Theorem

22.6 Jordan Decomposition

22.7 Nilpotent Linear Maps and Jordan Form

22.8 Summary

22.9 Problems

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Tags: Jean Gallier, Jocelyn Quaintance, linear algebra, Machine Learning