From Algebraic Structures to Tensors

Inbunden, Engelska, 2019

1 802 kr

Beställningsvara. Skickas inom 7-10 vardagar. Fri frakt över 249 kr.

Beskrivning

Nowadays, tensors play a central role for the representation, mining, analysis, and fusion of multidimensional, multimodal, and heterogeneous big data in numerous fields. This set on Matrices and Tensors in Signal Processing aims at giving a self-contained and comprehensive presentation of various concepts and methods, starting from fundamental algebraic structures to advanced tensor-based applications, including recently developed tensor models and efficient algorithms for dimensionality reduction and parameter estimation. Although its title suggests an orientation towards signal processing, the results presented in this set will also be of use to readers interested in other disciplines. This first book provides an introduction to matrices and tensors of higher-order based on the structures of vector space and tensor space. Some standard algebraic structures are first described, with a focus on the hilbertian approach for signal representation, and function approximation based on Fourier series and orthogonal polynomial series. Matrices and hypermatrices associated with linear, bilinear and multilinear maps are more particularly studied. Some basic results are presented for block matrices. The notions of decomposition, rank, eigenvalue, singular value, and unfolding of a tensor are introduced, by emphasizing similarities and differences between matrices and tensors of higher-order.

Produktinformation

Utgivningsdatum:2019-10-25
Mått:160 x 236 x 25 mm
Vikt:635 g
Format:Inbunden
Språk:Engelska
Antal sidor:320
Förlag:ISTE Ltd and John Wiley & Sons Inc
ISBN:9781786301543

Utforska kategorier

Systemvetenskap och AI inom Data och IT

Mer om författaren

Gérard Favier is currently Emeritus Research Director at CNRS and I3S Laboratory, in Sophia Antipolis, France. His research interests include nonlinear system modeling and identification, signal processing applications, tensor models with associated algorithms for big data processing, and tensor approaches for MIMO communication systems.

Innehållsförteckning

Preface xiChapter 1. Historical Elements of Matrices and Tensors 1Chapter 2. Algebraic Structures 92.1. A few historical elements 92.2. Chapter summary 112.3. Sets 122.3.1. Definitions 122.3.2. Sets of numbers 132.3.3. Cartesian product of sets 132.3.4. Set operations 142.3.5. De Morgan’s laws 152.3.6. Characteristic functions 152.3.7. Partitions 162.3.8. σ-algebras or σ-fields 162.3.9. Equivalence relations 162.3.10. Order relations 172.4. Maps and composition of maps 172.4.1. Definitions 172.4.2. Properties 182.4.3. Composition of maps 182.5. Algebraic structures 182.5.1. Laws of composition 182.5.2. Definition of algebraic structures 222.5.3. Substructures 242.5.4. Quotient structures 242.5.5. Groups 242.5.6. Rings 272.5.7. Fields 322.5.8. Modules 332.5.9. Vector spaces 332.5.10. Vector spaces of linear maps 382.5.11. Vector spaces of multilinear maps 392.5.12. Vector subspaces 412.5.13. Bases 432.5.14. Sum and direct sum of subspaces 452.5.15. Quotient vector spaces 472.5.16. Algebras 472.6. Morphisms 492.6.1. Group morphisms 492.6.2. Ring morphisms 512.6.3. Morphisms of vector spaces or linear maps 512.6.4. Algebra morphisms 56Chapter 3. Banach and Hilbert Spaces – Fourier Series and Orthogonal Polynomials 573.1. Introduction and chapter summary 573.2. Metric spaces 593.2.1. Definition of distance 603.2.2. Definition of topology 603.2.3. Examples of distances 613.2.4. Inequalities and equivalent distances 623.2.5. Distance and convergence of sequences 623.2.6. Distance and local continuity of a function 623.2.7. Isometries and Lipschitzian maps 633.3. Normed vector spaces 633.3.1. Definition of norm and triangle inequalities 633.3.2. Examples of norms 643.3.3. Equivalent norms 683.3.4. Distance associated with a norm 693.4. Pre-Hilbert spaces 693.4.1. Real pre-Hilbert spaces 703.4.2. Complex pre-Hilbert spaces 703.4.3. Norm induced from an inner product 723.4.4. Distance associated with an inner product 753.4.5. Weighted inner products 763.5. Orthogonality and orthonormal bases 763.5.1. Orthogonal/perpendicular vectors and Pythagorean theorem 763.5.2. Orthogonal subspaces and orthogonal complement 773.5.3. Orthonormal bases 793.5.4. Orthogonal/unitary endomorphisms and isometries 793.6. Gram–Schmidt orthonormalization process 803.6.1. Orthogonal projection onto a subspace 803.6.2. Orthogonal projection and Fourier expansion 803.6.3. Bessel’s inequality and Parseval’s equality 823.6.4. Gram–Schmidt orthonormalization process 833.6.5. QR decomposition 853.6.6. Application to the orthonormalization of a set of functions 863.7. Banach and Hilbert spaces 883.7.1. Complete metric spaces 883.7.2. Adherence, density and separability 903.7.3. Banach and Hilbert spaces 913.7.4. Hilbert bases 933.8. Fourier series expansions 973.8.1. Fourier series, Parseval’s equality and Bessel’s inequality 973.8.2. Case of 2π-periodic functions from R to C 973.8.3. T-periodic functions from R to C 1023.8.4. Partial Fourier sums and Bessel’s inequality 1023.8.5. Convergence of Fourier series 1033.8.6. Examples of Fourier series 1083.9. Expansions over bases of orthogonal polynomials 117Chapter 4. Matrix Algebra 1234.1. Chapter summary 1234.2. Matrix vector spaces 1244.2.1. Notations and definitions 1244.2.2. Partitioned matrices 1254.2.3. Matrix vector spaces 1264.3. Some special matrices 1274.4. Transposition and conjugate transposition 1284.5. Vectorization 1304.6. Vector inner product, norm and orthogonality 1304.6.1. Inner product 1304.6.2. Euclidean/Hermitian norm 1314.6.3. Orthogonality 1314.7. Matrix multiplication 1324.7.1. Definition and properties 1324.7.2. Powers of a matrix 1344.8. Matrix trace, inner product and Frobenius norm 1374.8.1. Definition and properties of the trace 1374.8.2. Matrix inner product 1384.8.3. Frobenius norm 1384.9. Subspaces associated with a matrix 1394.10. Matrix rank 1414.10.1. Definition and properties 1414.10.2. Sum and difference rank 1434.10.3. Subspaces associated with a matrix product 1434.10.4. Product rank 1444.11. Determinant, inverses and generalized inverses 1454.11.1. Determinant 1454.11.2. Matrix inversion 1484.11.3. Solution of a homogeneous system of linear equations 1494.11.4. Complex matrix inverse 1504.11.5. Orthogonal and unitary matrices 1504.11.6. Involutory matrices and anti-involutory matrices 1514.11.7. Left and right inverses of a rectangular matrix 1534.11.8. Generalized inverses 1554.11.9. Moore–Penrose pseudo-inverse 1574.12. Multiplicative groups of matrices 1584.13. Matrix associated to a linear map 1594.13.1. Matrix representation of a linear map 1594.13.2. Change of basis 1624.13.3. Endomorphisms 1644.13.4. Nilpotent endomorphisms 1664.13.5. Equivalent, similar and congruent matrices 1674.14. Matrix associated with a bilinear/sesquilinear form 1684.14.1. Definition of a bilinear/sesquilinear map 1684.14.2. Matrix associated to a bilinear/sesquilinear form 1704.14.3. Changes of bases with a bilinear form 1704.14.4. Changes of bases with a sesquilinear form 1714.14.5. Symmetric bilinear/sesquilinear forms 1724.15. Quadratic forms and Hermitian forms 1744.15.1. Quadratic forms 1744.15.2. Hermitian forms 1764.15.3. Positive/negative definite quadratic/Hermitian forms 1774.15.4. Examples of positive definite quadratic forms 1784.15.5. Cauchy–Schwarz and Minkowski inequalities 1794.15.6. Orthogonality, rank, kernel and degeneration of a bilinear form 1804.15.7. Gauss reduction method and Sylvester’s inertia law 1814.16. Eigenvalues and eigenvectors 1844.16.1. Characteristic polynomial and Cayley–Hamilton theorem 1844.16.2. Right eigenvectors 1864.16.3. Spectrum and regularity/singularity conditions 1874.16.4. Left eigenvectors 1884.16.5. Properties of eigenvectors 1884.16.6. Eigenvalues and eigenvectors of a regularized matrix 1904.16.7. Other properties of eigenvalues 1904.16.8. Symmetric/Hermitian matrices 1914.16.9. Orthogonal/unitary matrices 1934.16.10. Eigenvalues and extrema of the Rayleigh quotient 1944.17. Generalized eigenvalues 195Chapter 5. Partitioned Matrices 1995.1. Introduction 1995.2. Submatrices 2005.3. Partitioned matrices 2015.4. Matrix products and partitioned matrices 2025.4.1. Matrix products 2025.4.2. Vector Kronecker product 2025.4.3. Matrix Kronecker product 2025.4.4. Khatri–Rao product 2045.5. Special cases of partitioned matrices 2055.5.1. Block-diagonal matrices 2055.5.2. Signature matrices 2055.5.3. Direct sum 2055.5.4. Jordan forms 2065.5.5. Block-triangular matrices 2065.5.6. Block Toeplitz and Hankel matrices 2075.6. Transposition and conjugate transposition 2075.7. Trace 2085.8. Vectorization 2085.9. Blockwise addition 2085.10. Blockwise multiplication 2095.11. Hadamard product of partitioned matrices 2095.12. Kronecker product of partitioned matrices 2105.13. Elementary operations and elementary matrices 2125.14. Inversion of partitioned matrices 2145.14.1. Inversion of block-diagonal matrices 2155.14.2. Inversion of block-triangular matrices 2155.14.3. Block-triangularization and Schur complements 2165.14.4. Block-diagonalization and block-factorization 2165.14.5. Block-inversion and partitioned inverse 2175.14.6. Other formulae for the partitioned 2 × 2 inverse 2185.14.7. Solution of a system of linear equations 2195.14.8. Inversion of a partitioned Gram matrix 2205.14.9. Iterative inversion of a partitioned square matrix 2205.14.10. Matrix inversion lemma and applications 2215.15. Generalized inverses of 2 × 2 block matrices 2225.16. Determinants of partitioned matrices 2245.16.1. Determinant of block-diagonal matrices 2245.16.2. Determinant of block-triangular matrices 2255.16.3. Determinant of partitioned matrices with square diagonal blocks 2255.16.4. Determinants of specific partitioned matrices 2265.16.5. Eigenvalues of CB and BC 2275.17. Rank of partitioned matrices 2285.18. Levinson–Durbin algorithm 2295.18.1. AR process and Yule–Walker equations 2305.18.2. Levinson–Durbin algorithm 2325.18.3. Linear prediction 237Chapter 6. Tensor Spaces and Tensors 2436.1. Chapter summary 2436.2. Hypermatrices 2436.2.1. Hypermatrix vector spaces 2446.2.2. Hypermatrix inner product and Frobenius norm 2456.2.3. Contraction operation and n-mode hypermatrix–matrix product 2456.3. Outer products 2496.4. Multilinear forms, homogeneous polynomials and hypermatrices 2516.4.1. Hypermatrix associated to a multilinear form 2516.4.2. Symmetric multilinear forms and symmetric hypermatrices 2526.5. Multilinear maps and homogeneous polynomials 2556.6. Tensor spaces and tensors 2556.6.1. Definitions 2556.6.2. Multilinearity and associativity 2576.6.3. Tensors and coordinate hypermatrices 2576.6.4. Canonical writing of tensors 2586.6.5. Expansion of the tensor product of N vectors 2606.6.6. Properties of the tensor product 2616.6.7. Change of basis formula 2666.7. Tensor rank and tensor decompositions 2686.7.1. Matrix rank 2686.7.2. Hypermatrix rank 2686.7.3. Symmetric rank of a hypermatrix 2696.7.4. Comparative properties of hypermatrices and matrices 2696.7.5. CPD and dimensionality reduction 2716.7.6. Tensor rank 2736.8. Eigenvalues and singular values of a hypermatrix 2746.9. Isomorphisms of tensor spaces 276References 281Index 291