Muutke küpsiste eelistusi

E-raamat: From Algebraic Structures to Tensors

Edited by
  • Formaat: PDF+DRM
  • Ilmumisaeg: 05-Dec-2019
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119681090
Teised raamatud teemal:
  • Formaat - PDF+DRM
  • Hind: 171,60 €*
  • * hind on lõplik, st. muud allahindlused enam ei rakendu
  • Lisa ostukorvi
  • Lisa soovinimekirja
  • See e-raamat on mõeldud ainult isiklikuks kasutamiseks. E-raamatuid ei saa tagastada.
  • Raamatukogudele
From Algebraic Structures to TensorsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 05-Dec-2019
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119681090
Teised raamatud teemal:

DRM piirangud

  • Kopeerimine (copy/paste):

    ei ole lubatud

  • Printimine:

    ei ole lubatud

  • Kasutamine:

    Digitaalõiguste kaitse (DRM)
    Kirjastus on väljastanud selle e-raamatu krüpteeritud kujul, mis tähendab, et selle lugemiseks peate installeerima spetsiaalse tarkvara. Samuti peate looma endale  Adobe ID Rohkem infot siin. E-raamatut saab lugeda 1 kasutaja ning alla laadida kuni 6'de seadmesse (kõik autoriseeritud sama Adobe ID-ga).

    Vajalik tarkvara
    Mobiilsetes seadmetes (telefon või tahvelarvuti) lugemiseks peate installeerima selle tasuta rakenduse: PocketBook Reader (iOS / Android)

    PC või Mac seadmes lugemiseks peate installima Adobe Digital Editionsi (Seeon tasuta rakendus spetsiaalselt e-raamatute lugemiseks. Seda ei tohi segamini ajada Adober Reader'iga, mis tõenäoliselt on juba teie arvutisse installeeritud )

    Seda e-raamatut ei saa lugeda Amazon Kindle's. 

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.
Preface xi
Chapter 1 Historical Elements of Matrices and Tensors
1(8)
Chapter 2 Algebraic Structures
9(48)
2.1 A few historical elements
9(2)
2.2
Chapter summary
11(1)
2.3 Sets
12(5)
2.3.1 Definitions
12(1)
2.3.2 Sets of numbers
13(1)
2.3.3 Cartesian product of sets
13(1)
2.3.4 Set operations
14(1)
2.3.5 De Morgan's laws
15(1)
2.3.6 Characteristic functions
15(1)
2.3.7 Partitions
16(1)
2.3.8 σ-algebras or σ-fields
16(1)
2.3.9 Equivalence relations
16(1)
2.3.10 Order relations
17(1)
2.4 Maps and composition of maps
17(1)
2.4.1 Definitions
17(1)
2.4.2 Properties
18(1)
2.4.3 Composition of maps
18(1)
2.5 Algebraic structures
18(31)
2.5.1 Laws of composition
18(4)
2.5.2 Definition of algebraic structures
22(2)
2.5.3 Substructures
24(1)
2.5.4 Quotient structures
24(1)
2.5.5 Groups
24(3)
2.5.6 Rings
27(5)
2.5.7 Fields
32(1)
2.5.8 Modules
33(1)
2.5.9 Vector spaces
33(5)
2.5.10 Vector spaces of linear maps
38(1)
2.5.11 Vector spaces of multilinear maps
39(2)
2.5.12 Vector subspaces
41(2)
2.5.13 Bases
43(2)
2.5.14 Sum and direct sum of subspaces
45(2)
2.5.15 Quotient vector spaces
47(1)
2.5.16 Algebras
47(2)
2.6 Morphisms
49(8)
2.6.1 Group morphisms
49(2)
2.6.2 Ring morphisms
51(1)
2.6.3 Morphisms of vector spaces or linear maps
51(5)
2.6.4 Algebra morphisms
56(1)
Chapter 3 Banach and Hilbert Spaces - Fourier Series and Orthogonal Polynomials
57(66)
3.1 Introduction and chapter summary
57(2)
3.2 Metric spaces
59(4)
3.2.1 Definition of distance
60(1)
3.2.2 Definition of topology
60(1)
3.2.3 Examples of distances
61(1)
3.2.4 Inequalities and equivalent distances
62(1)
3.2.5 Distance and convergence of sequences
62(1)
3.2.6 Distance and local continuity of a function
62(1)
3.2.7 Isometries and Lipschitzian maps
63(1)
3.3 Normed vector spaces
63(6)
3.3.1 Definition of norm and triangle inequalities
63(1)
3.3.2 Examples of norms
64(4)
3.3.3 Equivalent norms
68(1)
3.3.4 Distance associated with a norm
69(1)
3.4 Pre-Hilbert spaces
69(7)
3.4.1 Real pre-Hilbert spaces
70(1)
3.4.2 Complex pre-Hilbert spaces
70(2)
3.4.3 Norm induced from an inner product
72(3)
3.4.4 Distance associated with an inner product
75(1)
3.4.5 Weighted inner products
76(1)
3.5 Orthogonality and orthonormal bases
76(4)
3.5.1 Orthogonal/perpendicular vectors and Pythagorean theorem
76(1)
3.5.2 Orthogonal subspaces and orthogonal complement
77(2)
3.5.3 Orthonormal bases
79(1)
3.5.4 Orthogonal/unitary endomorphisms and isometries
79(1)
3.6 Gram-Schmidt orthonormalization process
80(8)
3.6.1 Orthogonal projection onto a subspace
80(1)
3.6.2 Orthogonal projection and Fourier expansion
80(2)
3.6.3 Bessel's inequality and Parseval's equality
82(1)
3.6.4 Gram-Schmidt orthonormalization process
83(2)
3.6.5 QR decomposition
85(1)
3.6.6 Application to the orthonormalization of a set of functions
86(2)
3.7 Banach and Hilbert spaces
88(9)
3.7.1 Complete metric spaces
88(2)
3.7.2 Adherence, density and separability
90(1)
3.7.3 Banach and Hilbert spaces
91(2)
3.7.4 Hilbert bases
93(4)
3.8 Fourier series expansions
97(20)
3.8.1 Fourier series, Parseval's equality and Bessel's inequality
97(1)
3.8.2 Case of 2π-periodic functions from R to C
97(5)
3.8.3 T-periodic functions from R to C
102(1)
3.8.4 Partial Fourier sums and Bessel's inequality
102(1)
3.8.5 Convergence of Fourier series
103(5)
3.8.6 Examples of Fourier series
108(9)
3.9 Expansions over bases of orthogonal polynomials
117(6)
Chapter 4 Matrix Algebra
123(76)
4.1
Chapter summary
123(1)
4.2 Matrix vector spaces
124(3)
4.2.1 Notations and definitions
124(1)
4.2.2 Partitioned matrices
125(1)
4.2.3 Matrix vector spaces
126(1)
4.3 Some special matrices
127(1)
4.4 Transposition and conjugate transposition
128(2)
4.5 Vectorization
130(1)
4.6 Vector inner product, norm and orthogonality
130(2)
4.6.1 Inner product
130(1)
4.6.2 Euclidean/Hermitian norm
131(1)
4.6.3 Orthogonality
131(1)
4.7 Matrix multiplication
132(5)
4.7.1 Definition and properties
132(2)
4.7.2 Powers of a matrix
134(3)
4.8 Matrix trace, inner product and Frobenius norm
137(2)
4.8.1 Definition and properties of the trace
137(1)
4.8.2 Matrix inner product
138(1)
4.8.3 Frobenius norm
138(1)
4.9 Subspaces associated with a matrix
139(2)
4.10 Matrix rank
141(4)
4.10.1 Definition and properties
141(2)
4.10.2 Sum and difference rank
143(1)
4.10.3 Subspaces associated with a matrix product
143(1)
4.10.4 Product rank
144(1)
4.11 Determinant, inverses and generalized inverses
145(13)
4.11.1 Determinant
145(3)
4.11.2 Matrix inversion
148(1)
4.11.3 Solution of a homogeneous system of linear equations
149(1)
4.11.4 Complex matrix inverse
150(1)
4.11.5 Orthogonal and unitary matrices
150(1)
4.11.6 Involutory matrices and anti-involutory matrices
151(2)
4.11.7 Left and right inverses of a rectangular matrix
153(2)
4.11.8 Generalized inverses
155(2)
4.11.9 Moore-Penrose pseudo-inverse
157(1)
4.12 Multiplicative groups of matrices
158(1)
4.13 Matrix associated to a linear map
159(9)
4.13.1 Matrix representation of a linear map
159(3)
4.13.2 Change of basis
162(2)
4.13.3 Endomorphisms
164(2)
4.13.4 Nilpotent endomorphisms
166(1)
4.13.5 Equivalent, similar and congruent matrices
167(1)
4.14 Matrix associated with a bilinear/sesquilinear form
168(6)
4.14.1 Definition of a bilinear/sesquilinear map
168(2)
4.14.2 Matrix associated to a bilinear/sesquilinear form
170(1)
4.14.3 Changes of bases with a bilinear form
170(1)
4.14.4 Changes of bases with a sesquilinear form
171(1)
4.14.5 Symmetric bilinear/sesquilinear forms
172(2)
4.15 Quadratic forms and Hermitian forms
174(10)
4.15.1 Quadratic forms
174(2)
4.15.2 Hermitian forms
176(1)
4.15.3 Positive/negative definite quadratic/Hermitian forms
177(1)
4.15.4 Examples of positive definite quadratic forms
178(1)
4.15.5 Cauchy-Schwarz and Minkowski inequalities
179(1)
4.15.6 Orthogonality, rank, kernel and degeneration of a bilinear form
180(1)
4.15.7 Gauss reduction method and Sylvester's inertia law
181(3)
4.16 Eigenvalues and eigenvectors
184(11)
4.16.1 Characteristic polynomial and Cayley-Hamilton theorem
184(2)
4.16.2 Right eigenvectors
186(1)
4.16.3 Spectrum and regularity/singularity conditions
187(1)
4.16.4 Left eigenvectors
188(1)
4.16.5 Properties of eigenvectors
188(2)
4.16.6 Eigenvalues and eigenvectors of a regularized matrix
190(1)
4.16.7 Other properties of eigenvalues
190(1)
4.16.8 Symmetric/Hermitian matrices
191(2)
4.16.9 Orthogonal/unitary matrices
193(1)
4.16.10 Eigenvalues and extrema of the Rayleigh quotient
194(1)
4.17 Generalized eigenvalues
195(4)
Chapter 5 Partitioned Matrices
199(44)
5.1 Introduction
199(1)
5.2 Submatrices
200(1)
5.3 Partitioned matrices
201(1)
5.4 Matrix products and partitioned matrices
202(3)
5.4.1 Matrix products
202(1)
5.4.2 Vector Kronecker product
202(1)
5.4.3 Matrix Kronecker product
202(2)
5.4.4 Khatri-Rao product
204(1)
5.5 Special cases of partitioned matrices
205(2)
5.5.1 Block-diagonal matrices
205(1)
5.5.2 Signature matrices
205(1)
5.5.3 Direct sum
205(1)
5.5.4 Jordan forms
206(1)
5.5.5 Block-triangular matrices
206(1)
5.5.6 Block Toeplitz and Hankel matrices
207(1)
5.6 Transposition and conjugate transposition
207(1)
5.7 Trace
208(1)
5.8 Vectorization
208(1)
5.9 Blockwise addition
208(1)
5.10 Blockwise multiplication
209(1)
5.11 Hadamard product of partitioned matrices
209(1)
5.12 Kronecker product of partitioned matrices
210(2)
5.13 Elementary operations and elementary matrices
212(2)
5.14 Inversion of partitioned matrices
214(8)
5.14.1 Inversion of block-diagonal matrices
215(1)
5.14.2 Inversion of block-triangular matrices
215(1)
5.14.3 Block-triangularization and Schur complements
216(1)
5.14.4 Block-diagonalization and block-factorization
216(1)
5.14.5 Block-inversion and partitioned inverse
217(1)
5.14.6 Other formulae for the partitioned 2 × 2 inverse
218(1)
5.14.7 Solution of a system of linear equations
219(1)
5.14.8 Inversion of a partitioned Gram matrix
220(1)
5.14.9 Iterative inversion of a partitioned square matrix
220(1)
5.14.10 Matrix inversion lemma and applications
221(1)
5.15 Generalized inverses of 2 x 2 block matrices
222(2)
5.16 Determinants of partitioned matrices
224(4)
5.16.1 Determinant of block-diagonal matrices
224(1)
5.16.2 Determinant of block-triangular matrices
225(1)
5.16.3 Determinant of partitioned matrices with square diagonal blocks
225(1)
5.16.4 Determinants of specific partitioned matrices
226(1)
5.16.5 Eigenvalues of CB and BC
227(1)
5.17 Rank of partitioned matrices
228(1)
5.18 Levinson-Durbin algorithm
229(14)
5.18.1 AR process and Yule-Walker equations
230(2)
5.18.2 Levinson-Durbin algorithm
232(5)
5.18.3 Linear prediction
237(6)
Chapter 6 Tensor Spaces and Tensors
243(38)
6.1
Chapter summary
243(1)
6.2 Hypermatrices
243(6)
6.2.1 Hypermatrix vector spaces
244(1)
6.2.2 Hypermatrix inner product and Frobenius norm
245(1)
6.2.3 Contraction operation and n-mode hypermatrix-matrix product
245(4)
6.3 Outer products
249(2)
6.4 Multilinear forms, homogeneous polynomials and hypermatrices
251(4)
6.4.1 Hypermatrix associated to a multilinear form
251(1)
6.4.2 Symmetric multilinear forms and symmetric hypermatrices
252(3)
6.5 Multilinear maps and homogeneous polynomials
255(1)
6.6 Tensor spaces and tensors
255(13)
6.6.1 Definitions
255(2)
6.6.2 Multilinearity and associativity
257(1)
6.6.3 Tensors and coordinate hypermatrices
257(1)
6.6.4 Canonical writing of tensors
258(2)
6.6.5 Expansion of the tensor product of N vectors
260(1)
6.6.6 Properties of the tensor product
261(5)
6.6.7 Change of basis formula
266(2)
6.7 Tensor rank and tensor decompositions
268(6)
6.7.1 Matrix rank
268(1)
6.7.2 Hypermatrix rank
268(1)
6.7.3 Symmetric rank of a hypermatrix
269(1)
6.7.4 Comparative properties of hypermatrices and matrices
269(2)
6.7.5 CPD and dimensionality reduction
271(2)
6.7.6 Tensor rank
273(1)
6.8 Eigenvalues and singular values of a hypermatrix
274(2)
6.9 Isomorphisms of tensor spaces
276(5)
References 281(10)
Index 291
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.
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
Püsilink: https://www.kriso.ee/db/97811196810902e.html
Märksõnad: