From Algebraic Structures to Tensors [E-Book]

From Algebraic Structures to Tensors [E-Book]

The first volume will first present the basic elements of linear algebra and multilinear algebra. The notions of matrix spaces and tensor spaces will be introduced. Different examples of tensors encountered in signal processing and data analysis will be provided. Special classes of matrices and tens...

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Favier, Gérard, editor
Newark : John Wiley & Sons, Incorporated, 2020
1 online resource (323 pages)
englisch
9781119681113



  • Cover
  • Half-Title Page
  • Title Page
  • Copyright Page
  • Contents
  • Preface
  • 1. Historical Elements of Matrices and Tensors
  • 2. Algebraic Structures
  • 2.1. A few historical elements
  • 2.2. Chapter summary
  • 2.3. Sets
  • 2.3.1. Definitions
  • 2.3.2. Sets of numbers
  • 2.3.3. Cartesian product of sets
  • 2.3.4. Set operations
  • 2.3.5. De Morgan's laws
  • 2.3.6. Characteristic functions
  • 2.3.7. Partitions
  • 2.3.8. s-algebras or s-fields
  • 2.3.9. Equivalence relations
  • 2.3.10. Order relations
  • 2.4. Maps and composition of maps
  • 2.4.1. Definitions
  • 2.4.2. Properties
  • 2.4.3. Composition of maps
  • 2.5. Algebraic structures
  • 2.5.1. Laws of composition
  • 2.5.2. Definition of algebraic structures
  • 2.5.3. Substructures
  • 2.5.4. Quotient structures
  • 2.5.5. Groups
  • 2.5.6. Rings
  • 2.5.7. Fields
  • 2.5.8. Modules
  • 2.5.9. Vector spaces
  • 2.5.10. Vector spaces of linear maps
  • 2.5.11. Vector spaces of multilinear maps
  • 2.5.12. Vector subspaces
  • 2.5.13. Bases
  • 2.5.14. Sum and direct sum of subspaces
  • 2.5.15. Quotient vector spaces
  • 2.5.16. Algebras
  • 2.6. Morphisms
  • 2.6.1. Group morphisms
  • 2.6.2. Ring morphisms
  • 2.6.3. Morphisms of vector spaces or linear maps
  • 2.6.4. Algebra morphisms
  • 3. Banach and Hilbert Spaces - Fourier Series and Orthogonal Polynomials
  • 3.1. Introduction and chapter summary
  • 3.2. Metric spaces
  • 3.2.1. Definition of distance
  • 3.2.2. Definition of topology
  • 3.2.3. Examples of distances
  • 3.2.4. Inequalities and equivalent distances
  • 3.2.5. Distance and convergence of sequences
  • 3.2.6. Distance and local continuity of a function
  • 3.2.7. Isometries and Lipschitzian maps
  • 3.3. Normed vector spaces
  • 3.3.1. Definition of norm and triangle inequalities
  • 3.3.2. Examples of norms
  • 3.3.3. Equivalent norms
  • 3.3.4. Distance associated with a norm.
  • 3.4. Pre-Hilbert spaces
  • 3.4.1. Real pre-Hilbert spaces
  • 3.4.2. Complex pre-Hilbert spaces
  • 3.4.3. Norm induced from an inner product
  • 3.4.4. Distance associated with an inner product
  • 3.4.5. Weighted inner products
  • 3.5. Orthogonality and orthonormal bases
  • 3.5.1. Orthogonal/perpendicular vectors and Pythagorean theorem
  • 3.5.2. Orthogonal subspaces and orthogonal complement
  • 3.5.3. Orthonormal bases
  • 3.5.4. Orthogonal/unitary endomorphisms and isometries
  • 3.6. Gram-Schmidt orthonormalization process
  • 3.6.1. Orthogonal projection onto a subspace
  • 3.6.2. Orthogonal projection and Fourier expansion
  • 3.6.3. Bessel's inequality and Parseval's equality
  • 3.6.4. Gram-Schmidt orthonormalization process
  • 3.6.5. QR decomposition
  • 3.6.6. Application to the orthonormalization of a set of functions
  • 3.7. Banach and Hilbert spaces
  • 3.7.1. Complete metric spaces
  • 3.7.2. Adherence, density and separability
  • 3.7.3. Banach and Hilbert spaces
  • 3.7.4. Hilbert bases
  • 3.8. Fourier series expansions
  • 3.8.1. Fourier series, Parseval's equality and Bessel's inequality
  • 3.8.2. Case of 2p-periodic functions from R to C
  • 3.8.3. T-periodic functions from R to C
  • 3.8.4. Partial Fourier sums and Bessel's inequality
  • 3.8.5. Convergence of Fourier series
  • 3.8.6. Examples of Fourier series
  • 3.9. Expansions over bases of orthogonal polynomials
  • 4. Matrix Algebra
  • 4.1. Chapter summary
  • 4.2. Matrix vector spaces
  • 4.2.1. Notations and definitions
  • 4.2.2. Partitioned matrices
  • 4.2.3. Matrix vector spaces
  • 4.3. Some special matrices
  • 4.4. Transposition and conjugate transposition
  • 4.5. Vectorization
  • 4.6. Vector inner product, norm and orthogonality
  • 4.6.1. Inner product
  • 4.6.2. Euclidean/Hermitian norm
  • 4.6.3. Orthogonality
  • 4.7. Matrix multiplication
  • 4.7.1. Definition and properties.
  • 4.7.2. Powers of a matrix
  • 4.8. Matrix trace, inner product and Frobenius norm
  • 4.8.1. Definition and properties of the trace
  • 4.8.2. Matrix inner product
  • 4.8.3. Frobenius norm
  • 4.9. Subspaces associated with a matrix
  • 4.10. Matrix rank
  • 4.10.1. Definition and properties
  • 4.10.2. Sum and difference rank
  • 4.10.3. Subspaces associated with a matrix product
  • 4.10.4. Product rank
  • 4.11. Determinant, inverses and generalized inverses
  • 4.11.1. Determinant
  • 4.11.2. Matrix inversion
  • 4.11.3. Solution of a homogeneous system of linear equations
  • 4.11.4. Complex matrix inverse
  • 4.11.5. Orthogonal and unitary matrices
  • 4.11.6. Involutory matrices and anti-involutory matrices
  • 4.11.7. Left and right inverses of a rectangular matrix
  • 4.11.8. Generalized inverses
  • 4.11.9. Moore-Penrose pseudo-inverse
  • 4.12. Multiplicative groups of matrices
  • 4.13. Matrix associated to a linear map
  • 4.13.1. Matrix representation of a linear map
  • 4.13.2. Change of basis
  • 4.13.3. Endomorphisms
  • 4.13.4. Nilpotent endomorphisms
  • 4.13.5. Equivalent, similar and congruent matrices
  • 4.14. Matrix associated with a bilinear/sesquilinear form
  • 4.14.1. Definition of a bilinear/sesquilinear map
  • 4.14.2. Matrix associated to a bilinear/sesquilinear form
  • 4.14.3. Changes of bases with a bilinear form
  • 4.14.4. Changes of bases with a sesquilinear form
  • 4.14.5. Symmetric bilinear/sesquilinear forms
  • 4.15. Quadratic forms and Hermitian forms
  • 4.15.1. Quadratic forms
  • 4.15.2. Hermitian forms
  • 4.15.3. Positive/negative definite quadratic/Hermitian forms
  • 4.15.4. Examples of positive definite quadratic forms
  • 4.15.5. Cauchy-Schwarz and Minkowski inequalities
  • 4.15.6. Orthogonality, rank, kernel and degeneration of a bilinear form
  • 4.15.7. Gauss reduction method and Sylvester's inertia law.
  • 4.16. Eigenvalues and eigenvectors
  • 4.16.1. Characteristic polynomial and Cayley-Hamilton theorem
  • 4.16.2. Right eigenvectors
  • 4.16.3. Spectrum and regularity/singularity conditions
  • 4.16.4. Left eigenvectors
  • 4.16.5. Properties of eigenvectors
  • 4.16.6. Eigenvalues and eigenvectors of a regularized matrix
  • 4.16.7. Other properties of eigenvalues
  • 4.16.8. Symmetric/Hermitian matrices
  • 4.16.9. Orthogonal/unitary matrices
  • 4.16.10. Eigenvalues and extrema of the Rayleigh quotient
  • 4.17. Generalized eigenvalues
  • 5. Partitioned Matrices
  • 5.1. Introduction
  • 5.2. Submatrices
  • 5.3. Partitioned matrices
  • 5.4. Matrix products and partitioned matrices
  • 5.4.1. Matrix products
  • 5.4.2. Vector Kronecker product
  • 5.4.3. Matrix Kronecker product
  • 5.4.4. Khatri-Rao product
  • 5.5. Special cases of partitioned matrices
  • 5.5.1. Block-diagonal matrices
  • 5.5.2. Signature matrices
  • 5.5.3. Direct sum
  • 5.5.4. Jordan forms
  • 5.5.5. Block-triangular matrices
  • 5.5.6. Block Toeplitz and Hankel matrices
  • 5.6. Transposition and conjugate transposition
  • 5.7. Trace
  • 5.8. Vectorization
  • 5.9. Blockwise addition
  • 5.10. Blockwise multiplication
  • 5.11. Hadamard product of partitioned matrices
  • 5.12. Kronecker product of partitioned matrices
  • 5.13. Elementary operations and elementary matrices
  • 5.14. Inversion of partitioned matrices
  • 5.14.1. Inversion of block-diagonal matrices
  • 5.14.2. Inversion of block-triangular matrices
  • 5.14.3. Block-triangularization and Schur complements
  • 5.14.4. Block-diagonalization and block-factorization
  • 5.14.5. Block-inversion and partitioned inverse
  • 5.14.6. Other formulae for the partitioned 2 × 2 inverse
  • 5.14.7. Solution of a system of linear equations
  • 5.14.8. Inversion of a partitioned Gram matrix
  • 5.14.9. Iterative inversion of a partitioned square matrix.
  • 5.14.10. Matrix inversion lemma and applications
  • 5.15. Generalized inverses of 2 × 2 block matrices
  • 5.16. Determinants of partitioned matrices
  • 5.16.1. Determinant of block-diagonal matrices
  • 5.16.2. Determinant of block-triangular matrices
  • 5.16.3. Determinant of partitioned matrices with square diagonal blocks
  • 5.16.4. Determinants of specific partitioned matrices
  • 5.16.5. Eigenvalues of CB and BC
  • 5.17. Rank of partitioned matrices
  • 5.18. Levinson-Durbin algorithm
  • 5.18.1. AR process and Yule-Walker equations
  • 5.18.2. Levinson-Durbin algorithm
  • 5.18.3. Linear prediction
  • 6. Tensor Spaces and Tensors
  • 6.1. Chapter summary
  • 6.2. Hypermatrices
  • 6.2.1. Hypermatrix vector spaces
  • 6.2.2. Hypermatrix inner product and Frobenius norm
  • 6.2.3. Contraction operation and n-mode hypermatrix-matrix product
  • 6.3. Outer products
  • 6.4. Multilinear forms, homogeneous polynomials and hypermatrices
  • 6.4.1. Hypermatrix associated to a multilinear form
  • 6.4.2. Symmetric multilinear forms and symmetric hypermatrices
  • 6.5. Multilinear maps and homogeneous polynomials
  • 6.6. Tensor spaces and tensors
  • 6.6.1. Definitions
  • 6.6.2. Multilinearity and associativity
  • 6.6.3. Tensors and coordinate hypermatrices
  • 6.6.4. Canonical writing of tensors
  • 6.6.5. Expansion of the tensor product of N vectors
  • 6.6.6. Properties of the tensor product
  • 6.6.7. Change of basis formula
  • 6.7. Tensor rank and tensor decompositions
  • 6.7.1. Matrix rank
  • 6.7.2. Hypermatrix rank
  • 6.7.3. Symmetric rank of a hypermatrix
  • 6.7.4. Comparative properties of hypermatrices and matrices
  • 6.7.5. CPD and dimensionality reduction
  • 6.7.6. Tensor rank
  • 6.8. Eigenvalues and singular values of a hypermatrix
  • 6.9. Isomorphisms of tensor spaces
  • References
  • Index
  • Other titles from iISTE in Digital Signal and Image Processing.
  • EULA.