Jacobians of matrix transformations and functions of matrix argument:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
1997
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 435 S. |
| ISBN: | 9810230958 |
Staff View
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| 245 | 1 | 0 | |a Jacobians of matrix transformations and functions of matrix argument |c A. M. Mathai |
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| 300 | |a XII, 435 S. | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
1. Jacobians of matrix transformations 1
1.0 Introduction 1
1.1 Vector and matrix derivatives 2
1.1.1 The derivative of a matrix with respect to a scalar 2
1.1.2 The derivative of a scalar function of a vector
with respect to the vector 2
1.1.3 The derivative of a scalar function of a matrix
with respect to the matrix 6
1.1.4 The derivative of a matrix with respect to
an element of the same matrix 14
Exercises 19
1.2 Jacobians of linear matrix transformations 21
1.2.1 Jacobians of vector transformations 21
1.2.2 Linear matrix transformations 24
1.2.3 Function of a function, linear case 31
Exercises 39
1.3 Nonlinear transformations: one matrix case 43
1.3.1 Nonlinear vector transformations 43
1.3.2 Smooth nonlinear transformations 47
1.3.3 Function of a function 60
1.3.4 Jacobians in terms of eigenvalues 66
Exercises 78
2. Jacobians in orthogonal and related transformations 83
2.0 Introduction 83
2.1 Transformations of one matrix to components of two
matrices 83
2.1.1 Decomposition with one matrix being triangular
or diagonal 83
2.1.2 Decomposition with one matrix being skew
symmetric 99
Exercises 111
2.2 Decomposition with one matrix being orthogonal or
and element of the Stiefel manifold 115
ix
X
Exercises 135
2.3 Singular value decomposition 137
2.3.1 Square matrices of full rank 137
2.3.2 Rectangular matrices of full rank 142
Exercises 145
2.4 Simultaneous transformations involving many matrices 147
2.4.1 Transformations involving two matrices 147
2.4.2 Transformations involving many matrices 153
Exercises 158
2.5 Transformations involving submatrices 159
Exercises 169
3. Jacobians in the complex case 171
3.0 Introduction 171
3.1 Jacobians of linear transformations in the complex case 175
Exercises 185
3.2 Smooth nonlinear transformations in the complex case 187
Exercises 198
3.3 Transformations involving determinants and diagonal
matrices 199
Exercises 219
4. Transformations involving eigenvalues and unitary
matrices 221
4.0 Introduction 221
4.1 Transformations involving skew hermitian matrices 221
4.1.1 A direct evaluation of the integral 228
Exercises 231
4.2 Transformations involving unitary matrices 232
4.2.1 Semiunitary transformations 240
Exercises 243
4.3 Transformations involving singular values 245
4.3.1 Square matrices of full rank 245
4.3.2 Rectangular matrices of full rank 250
Exercises 252
5. Some special functions of matrix argument 253
5.0 Introduction 253
5.1 Elementary functions of matrix argument 253
5.1.1 Matrix variate real gamma density 254
xi
5.1.2 Laplace transform 254
5.1.3 Functions of matrix argument through
Laplace transform 257
5.1.4 Matrix variate real beta density 259
5.1.5 Functions of matrix argument through zonal
polynomials 265
5.1.6 The Weyl fractional integral and fractional
derivative 267
5.1.7 The M transform 271
5.1.8 Matrix variate Dirichlet integrals and Dirichlet
distributions 274
5.1.9 Real matrix variate Liouville function and
Liouville distribution 278
5.1.10 Some applications 279
Exercises 280
5.2 Hypergeometric functions of matrix argument 282
5.2.1 Hypergeometric functions of matrix argument
through Laplace transform 283
5.2.2 Hypergeometric functions through zonal
polynomials 288
5.2.3 Bessel function of matrix argument 291
5.2.4 Hypergeometric function of matrix argument
through M transform 294
5.2.5 Whittaker function of matrix argument 297
5.2.6 Meijer s G and Fox s H functions of matrix
argument through M transform 301
5.2.7 Some applications 303
Exercises 304
5.3 Appell s and Humbert s functions 307
5.3.1 Appell s functions of matrix arguments 308
5.3.2 Humbert s functions of matrix arguments 313
5.3.3 Humbert s functions as limiting forms 317
Exercises 321
5.4 Kampi de Feiiet s functions of matrix arguments 322
5.4.1 Definition 322
5.4.2 Some integral representations 323
5.4.3 Some limiting forms 328
5.4.4 Some reduction formulae 330
5.4.5 An application in astrophysics problems 331
Exercises 332
5.5 Lauricella functions of matrix arguments 333
xii
5.5.1 Lauricella function /a of matrix arguments 333
5.5.2 Lauricella function fs of matrix arguments 337
5.5.3 Lauricella function fc of matrix arguments 342
5.5.4 Lauricella function fo of matrix arguments 343
5.5.5 Some mixed results 345
5.5.6 Some applications to statistical distributions 347
(a) Distribution of a linear form 347
(b) Distribution of a determinant 352
(c) Generalized quadratic forms 353
Exercises 354
6. Functions of matrix argument in the complex case 355
6.0 Introduction 355
6.1 Some elementary functions in the complex case 356
6.1.1 Laplace transform in the complex case 358
6.1.2 Zonal polynomials in the complex case 365
6.1.3 The M transform 367
Exercises 368
6.2 Hypergeometric functions of matrix argument:
complex case 369
Exercises 373
6.3 G and H functions of matrix argument:
complex case 375
Exercises 380
6.4 Functions of several matrix arguments:
complex case 381
6.4.1 Dirichlet functions: complex case 381
6.4.2 Appell s functions: complex case 385
6.4.3 Series forms: complex case 389
6.4.4 Humbert s functions: complex case 390
6.4.5 Kampe de Fe riet s functions: complex case 394
6.4.6 Lauricella functions: complex case 396
Exercises 399
Appendix 401
Bibliography 411
Glossary of symbols 421
Author index 425
Subject index 429
|
| any_adam_object | 1 |
| author | Mathai, Arakaparampil M. 1935- |
| author_GND | (DE-588)141316241 |
| author_facet | Mathai, Arakaparampil M. 1935- |
| author_role | aut |
| author_sort | Mathai, Arakaparampil M. 1935- |
| author_variant | a m m am amm |
| building | Verbundindex |
| bvnumber | BV012731781 |
| callnumber-first | Q - Science |
| callnumber-label | QA191 |
| callnumber-raw | QA191.M36 1997 |
| callnumber-search | QA191.M36 1997 |
| callnumber-sort | QA 3191 M36 41997 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 220 SK 640 |
| ctrlnum | (OCoLC)36343319 (DE-599)BVBBV012731781 |
| dewey-full | 512.9/434 512.9/43421 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/434 512.9/434 21 |
| dewey-search | 512.9/434 512.9/434 21 |
| dewey-sort | 3512.9 3434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T18:32:43Z |
| institution | BVB |
| isbn | 9810230958 |
| language | English |
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| physical | XII, 435 S. |
| publishDate | 1997 |
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| spelling | Mathai, Arakaparampil M. 1935- Verfasser (DE-588)141316241 aut Jacobians of matrix transformations and functions of matrix argument A. M. Mathai Singapore [u.a.] World Scientific 1997 XII, 435 S. txt rdacontent n rdamedia nc rdacarrier Fonctions hypergéométriques ram Fonctions spéciales ram Jacobi-functies gtt Jacobiens ram Matrices - Problèmes et exercices ram Matrices gtt Matrices ram Transformaties (wiskunde) gtt Transformations (mathématiques) ram fonction hypergéométrique inriac fonction spéciale inriac jacobien inriac matrice inriac transformation inriac Jacobians Matrices Transformations (Mathematics) Matrixtransformation (DE-588)4572058-7 gnd rswk-swf Funktionaldeterminante (DE-588)4417067-1 gnd rswk-swf Funktionaldeterminante (DE-588)4417067-1 s Matrixtransformation (DE-588)4572058-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008656761&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mathai, Arakaparampil M. 1935- Jacobians of matrix transformations and functions of matrix argument Fonctions hypergéométriques ram Fonctions spéciales ram Jacobi-functies gtt Jacobiens ram Matrices - Problèmes et exercices ram Matrices gtt Matrices ram Transformaties (wiskunde) gtt Transformations (mathématiques) ram fonction hypergéométrique inriac fonction spéciale inriac jacobien inriac matrice inriac transformation inriac Jacobians Matrices Transformations (Mathematics) Matrixtransformation (DE-588)4572058-7 gnd Funktionaldeterminante (DE-588)4417067-1 gnd |
| subject_GND | (DE-588)4572058-7 (DE-588)4417067-1 |
| title | Jacobians of matrix transformations and functions of matrix argument |
| title_auth | Jacobians of matrix transformations and functions of matrix argument |
| title_exact_search | Jacobians of matrix transformations and functions of matrix argument |
| title_full | Jacobians of matrix transformations and functions of matrix argument A. M. Mathai |
| title_fullStr | Jacobians of matrix transformations and functions of matrix argument A. M. Mathai |
| title_full_unstemmed | Jacobians of matrix transformations and functions of matrix argument A. M. Mathai |
| title_short | Jacobians of matrix transformations and functions of matrix argument |
| title_sort | jacobians of matrix transformations and functions of matrix argument |
| topic | Fonctions hypergéométriques ram Fonctions spéciales ram Jacobi-functies gtt Jacobiens ram Matrices - Problèmes et exercices ram Matrices gtt Matrices ram Transformaties (wiskunde) gtt Transformations (mathématiques) ram fonction hypergéométrique inriac fonction spéciale inriac jacobien inriac matrice inriac transformation inriac Jacobians Matrices Transformations (Mathematics) Matrixtransformation (DE-588)4572058-7 gnd Funktionaldeterminante (DE-588)4417067-1 gnd |
| topic_facet | Fonctions hypergéométriques Fonctions spéciales Jacobi-functies Jacobiens Matrices - Problèmes et exercices Matrices Transformaties (wiskunde) Transformations (mathématiques) fonction hypergéométrique fonction spéciale jacobien matrice transformation Jacobians Transformations (Mathematics) Matrixtransformation Funktionaldeterminante |
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