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EB

ONLINE

Newark : John Wiley & Sons, Incorporated, 2020

1 online resource (323 pages)

ISBN 9781119681113 (electronic bk.)

ISBN 9781786301543

Print version: Favier, Gerard From Algebraic Structures to Tensors Newark : John Wiley & Sons, Incorporated,c2020 ISBN 9781786301543

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.

EULA.

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.

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