Linear Algebra Tools for Data Mining

Contents
Preface
About the Author
1. Preliminaries
1.1 Introduction
1.2 Functions
1.3 Sequences
1.4 Permutations
1.5 Combinatorics
1.6 Groups, Rings, and Fields
1.7 Closure and Interior Systems
Bibliographical Comments
2. Linear Spaces
2.1 Introduction
2.2 Linear Spaces
2.3 Linear Independence
2.4 Linear Mappings
2.5 Bases in Linear Spaces
2.6 Isomorphisms of Linear Spaces
2.7 Constructing Linear Spaces
2.8 Dual Linear Spaces
2.9 Topological Linear Spaces
2.10 Isomorphism Theorems
2.11 Multilinear Functions
Exercises and Supplements
Bibliographical Comments
3. Matrices
3.1 Introduction
3.2 Matrices with Arbitrary Elements
3.3 Fields and Matrices
3.4 Invertible Matrices
3.5 Special Classes of Matrices
3.6 Partitioned Matrices and Matrix Operations
3.7 Change of Bases
3.8 Matrices and Bilinear Forms
3.9 Generalized Inverses of Matrices
3.10 Matrices and Linear Transformations
3.11 The Notion of Rank
3.12 Matrix Similarity and Congruence
3.13 Linear Systems and LU Decompositions
3.14 The Row Echelon Form of Matrices
3.15 The Kronecker and Other Matrix Products
3.16 Outer Products
3.17 Associative Algebras
Exercises and Supplements
Bibliographical Comments
4. MATLAB Environment
4.1 Introduction
4.2 The Interactive Environment of MATLAB
4.3 Number Representation and Arithmetic Computations
4.4 Matrices and Multidimensional Arrays
4.5 Cell Arrays
4.6 Solving Linear Systems
4.7 Control Structures
4.8 Indexing
4.9 Functions
4.10 Matrix Computations
4.11 Matrices and Images in MATLAB
Exercises and Supplements
Bibliographical Comments
5. Determinants
5.1 Introduction
5.2 Determinants and Multilinear Forms
5.3 Cramer’s Formula
5.4 Partitioned Matrices and Determinants
5.5 Resultants
5.6 MATLAB Computations
Exercises and Supplements
Bibliographical Comments
6. Norms and Inner Products
6.1 Introduction
6.2 Basic Inequalities
6.3 Metric Spaces
6.4 Norms
6.5 The Topology of Normed Linear Spaces
6.6 Norms for Matrices
6.7 Matrix Sequences and Matrix Series
6.8 Conjugate Norms
6.9 Inner Products
6.10 Hyperplanes in Rn
6.11 Unitary and Orthogonal Matrices
6.12 Projection on Subspaces
6.13 Positive Definite and Positive Semidefinite Matrices
6.14 The Gram–Schmidt Orthogonalization Algorithm
6.15 Change of Bases Revisited
6.16 The QR Factorization of Matrices
6.17 Matrix Groups
6.18 Condition Numbers for Matrices
6.19 Linear Space Orientation
6.20 MATLAB Computations
Exercises and Supplements
Bibliographical Comments
7. Eigenvalues
7.1 Introduction
7.2 Eigenvalues and Eigenvectors
7.3 The Characteristic Polynomial of a Matrix
7.4 Spectra of Hermitian Matrices
7.5 Spectra of Special Matrices
7.6 Geometry of Eigenvalues
7.7 Spectra of Kronecker Products and Sums
7.8 The Power Method for Eigenvalues
7.9 The QR Iterative Algorithm
7.10 MATLAB Computations
Exercises and Supplements
Bibliographical Comments
8. Similarity and Spectra
8.1 Introduction
8.2 Diagonalizable Matrices
8.3 Matrix Similarity and Spectra
8.4 The Sylvester Operator
8.5 Geometric versus Algebraic Multiplicity
8.6 λ-Matrices
8.7 The Jordan Canonical Form
8.8 Matrix Norms and Eigenvalues
8.9 Matrix Pencils and Generalized Eigenvalues
8.10 Quadratic Forms and Quadrics
8.11 Spectra of Positive Matrices
8.12 Spectra of Positive Semidefinite Matrices
8.13 MATLAB Computations
Exercises and Supplements
Bibliographical Comments
9. Singular Values
9.1 Introduction
9.2 Singular Values and Singular Vectors
9.3 Numerical Rank of Matrices
9.4 Updating SVDs
9.5 Polar Form of Matrices
9.6 CS Decomposition
9.7 Geometry of Subspaces
9.8 Spectral Resolution of a Matrix
9.9 MATLAB Computations
Exercises and Supplements
Bibliographical Comments
10. The k-Means Clustering
10.1 Introduction
10.2 The k-Means Algorithm and Convexity
10.3 Relaxation of the k-Means Problem
10.4 SVD and Clustering
10.5 Evaluation of Clusterings
10.6 MATLAB Computations
Exercises and Supplements
Bibliographical Comments
11. Data Sample Matrices
11.1 Introduction
11.2 The Sample Matrix
11.3 Biplots
Exercises and Supplements
Bibliographical Comments
12. Least Squares Approximations and Data Mining
12.1 Introduction
12.2 Linear Regression
12.3 The Least Square Approximation and QR Decomposition
12.4 Partial Least Square Regression
12.5 Locally Linear Embedding
12.6 MATLAB Computations
Exercises and Supplements
Bibliographical Comments
13. Dimensionality Reduction Techniques
13.1 Introduction
13.2 Principal Component Analysis
13.3 Linear Discriminant Analysis
13.4 Latent Semantic Indexing
13.5 Recommender Systems and SVD
13.6 Metric Multidimensional Scaling
13.7 Procrustes Analysis
13.8 Non-negative Matrix Factorization
Exercises and Supplements
Bibliographical Comments
14. Tensors and Exterior Algebras
14.1 Introduction
14.2 The Summation Convention
14.3 Tensor Products of Linear Spaces
14.4 Tensors on Inner Product Spaces
14.5 Contractions
14.6 Symmetric and Skew-Symmetric Tensors
14.7 Exterior Algebras
14.8 Linear Mappings between Spaces SKSV,k
14.9 Determinants and Exterior Algebra
Exercises and Supplements
Bibliographical Comments
15. Multidimensional Array and Tensors
15.1 Introduction
15.2 Multidimensional Arrays
15.3 Outer Products
15.4 Tensor Rank
15.5 Matricization and Vectorization
15.6 Inner Product and Norms
15.7 Evaluation of a Set of Bilinear Forms
15.8 Matrix Multiplications and Arrays
15.9 MATLAB Computations
15.10 Hyperdeterminants
15.11 Eigenvalues and Singular Values
15.12 Decomposition of Tensors
15.13 Approximation of mdas
Exercises and Supplements
Bibliographical Comments
Bibliography
Index