Advanced linear and matrix algebra:
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| Format: | Book |
| Language: | English |
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Cham
Springer
[2021]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvi, 494 Seiten Illustrationen, Diagramme |
| ISBN: | 9783030528140 |
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Record in the Search Index
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| adam_text | Preface............................................................................................................ vii The Purpose of this Book...................................................................................... vii Features of this Book............................................................................................ viii To the Instructor and Independent Reader.................................................... viii Chapter 1: Vector Spaces......................................................................... 1 Vector Spaces and Subspaces............................................................................ 2 1.1 8 1.1.1 Subspaces 1.1.2 Spans, Linear Combinations, and Independence 1.1.3 Bases 18 Exercises 22 1.2 Coordinates and Linear Transformations............................................................ 11 24 1.2.1 Dimension and Coordinate Vectors 24 1.2.2 Change of Basis 30 1.2.3 Linear Transformations 36 1.2.4 Properties of Linear Transformations 44 Exercises 51 1.3 Isomorphisms and Linear Forms............................................................................ 54 1.3.1 Isomorphisms 54 1.3.2 Linear Forms 58 1.3.3 Bilinearity and Beyond 64 1.3.4 Inner Products 70 Exercises 77 1.4 Orthogonality and Adjoints.................................................................................... 79 1.4.1 Orthonormal Bases 81 1.4,2 Adjoint Transformations 92 xiii
Table of Contents xiv 1.4.3 Unitary Matrices 1,4.4 Projections 101 Exercises 112 Summary and Review............................................................................................. 1 .5 96 114 l .A Extra Topic: More About the Trace................................................................... 116 Extra Topic: Direct Sum, Orthogonal Complement.......................................... 121 l .C Extra Topic: The QR Decomposition................................................................... 139 Extra Topic: Norms and Isometries........................................................................ 146 l.B l.D Chapter 2: Matrix Decompositions 2.1 167 The Schur and Spectral Decompositions............................................................ 168 2.1.1 Schur Triangularization 169 2.1.2 Normal Matrices and the Complex Spectral Decomposition 176 2.1.3 The Real Spectral Decomposition 182 Exercises 186 2.2 Positive Semidefiniteness........................................................................................ 188 2.2.1 Characterizing Positive (Semi)Definite Matrices 189 2.2.2 Diagonal Dominance and Gershgorin Discs 197 2.2.3 Unitary Freedom of PSD Decompositions 201 Exercises 207 2.3 The Singular Value Decomposition..................................................................... 209 2.3.1 Geometric Interpretation and the Fundamental Subspaces 212 2.3.2 Relationship with Other Matrix Decompositions 217 2.3.3 The Operator Norm 220 Exercises 224 2.4 The Jordan
Decomposition................................................................................... 226 2.4.1 Uniqueness and Similarity 228 2.4.2 Existence and Computation 234 2.4.3 Matrix Functions 245 Exercises 251 2.5 Summary and Review............................................................................................. 252 2.A Extra Topic: Quadratic Forms and ConicSections............................................. 255 2.В Extra Topic: Schur Complements and Cholesky.............................................. 263 2.C Extra Topic: Applications of the SVD.................................................................... 273 2.D Extra Topic: Continuity and Matrix Analysis....................................................... 284
Table of Contents Chapter 3: Tensors and Multilinearity The Kronecker Product............................................................................................ 3.1 xv 297 297 3.1.1 Definition and Basic Properties 298 3.1.2 Vectorization and the Swap Matrix 305 3.1.3 The Symmetric and Antisymmetric Subspaces 311 Exercises 318 3.2 Multilinear Transformations................................................................................... 320 3.2.1 Definition and Basic Examples 320 3.2.2 Arrays 323 3.2.3 Properties of Multilinear Transformations 333 Exercises 340 The Tensor Product................................................................................................. 342 3.3 3.3.1 Motivation and Definition 342 3.3.2 Existence and Uniqueness 345 3.3.3 Tensor Rank 350 Exercises 356 3.4 Summary and Review............................................................................................. 357 3.A Extra Topic: Matrix-Valued Linear Maps............................................................. 358 3.B Extra Topic: Homogeneous Polynomials........................................................... 377 3.C Extra Topic: Semidefinite Programming............................................................. 393 Appendix A: Mathematical Preliminaries A.l Review of Introductory Linear Algebra............................................................... 415 415 A. 1.1 Systems of Linear Equations 415 A.1.2 Matrices as Linear Transformations 416 A. 1.3 The Inverse of a Matrix 417 A. 1.4 Range, Rank, Null Space, and Nullity 418 A. 1.5 Determinants and
Permutations 419 A. 1.6 Eigenvalues and Eigenvectors 421 A. 1.7 Diagonalization 422 A.2 Polynomials and Beyond........................................................................................ 424 A.2.1 Monomials, Binomials and Multinomials 424 A.2.2 Taylor Polynomials and Taylor Series 426
Table of Contents xvi A.3 Complex Numbers............................................................................................. 428 A.3.1 Basic Arithmetic and Geometry 429 A.3.2 The Complex Conjugate 430 A.3.3 Euler’s Formula and Polar Form 431 A.4 Fields................................................................................................................... A.5 Convexity............................................................................................................ 436 434 A.5.1 Convex Sets 436 A.5.2 Convex Functions 438 Appendix B: Additional Proofs 441 B.l Equivalence of Norms..................................................................................... B.2 Details of the Jordan Decomposition........................................................... 442 B.3 Strong Duality for Semidefinite Programs..................................................... 446 Appendix C: Selected Exercise Solutions 441 449 C.l Chapter 1 : Vector Spaces............................................................................ 451 C.2 Chapter 2: Matrix Decompositions............................................................... 465 C.3 Chapter 3: Tensors and Multilinearity........................................................... 476 Bibliography.................................................................................................................. 487 Index............................................................................................................................. 489 Symbol
Index................................................................................................................ 494
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| adam_txt |
Preface. vii The Purpose of this Book. vii Features of this Book. viii To the Instructor and Independent Reader. viii Chapter 1: Vector Spaces. 1 Vector Spaces and Subspaces. 2 1.1 8 1.1.1 Subspaces 1.1.2 Spans, Linear Combinations, and Independence 1.1.3 Bases 18 Exercises 22 1.2 Coordinates and Linear Transformations. 11 24 1.2.1 Dimension and Coordinate Vectors 24 1.2.2 Change of Basis 30 1.2.3 Linear Transformations 36 1.2.4 Properties of Linear Transformations 44 Exercises 51 1.3 Isomorphisms and Linear Forms. 54 1.3.1 Isomorphisms 54 1.3.2 Linear Forms 58 1.3.3 Bilinearity and Beyond 64 1.3.4 Inner Products 70 Exercises 77 1.4 Orthogonality and Adjoints. 79 1.4.1 Orthonormal Bases 81 1.4,2 Adjoint Transformations 92 xiii
Table of Contents xiv 1.4.3 Unitary Matrices 1,4.4 Projections 101 Exercises 112 Summary and Review. 1 .5 96 114 l .A Extra Topic: More About the Trace. 116 Extra Topic: Direct Sum, Orthogonal Complement. 121 l .C Extra Topic: The QR Decomposition. 139 Extra Topic: Norms and Isometries. 146 l.B l.D Chapter 2: Matrix Decompositions 2.1 167 The Schur and Spectral Decompositions. 168 2.1.1 Schur Triangularization 169 2.1.2 Normal Matrices and the Complex Spectral Decomposition 176 2.1.3 The Real Spectral Decomposition 182 Exercises 186 2.2 Positive Semidefiniteness. 188 2.2.1 Characterizing Positive (Semi)Definite Matrices 189 2.2.2 Diagonal Dominance and Gershgorin Discs 197 2.2.3 Unitary Freedom of PSD Decompositions 201 Exercises 207 2.3 The Singular Value Decomposition. 209 2.3.1 Geometric Interpretation and the Fundamental Subspaces 212 2.3.2 Relationship with Other Matrix Decompositions 217 2.3.3 The Operator Norm 220 Exercises 224 2.4 The Jordan
Decomposition. 226 2.4.1 Uniqueness and Similarity 228 2.4.2 Existence and Computation 234 2.4.3 Matrix Functions 245 Exercises 251 2.5 Summary and Review. 252 2.A Extra Topic: Quadratic Forms and ConicSections. 255 2.В Extra Topic: Schur Complements and Cholesky. 263 2.C Extra Topic: Applications of the SVD. 273 2.D Extra Topic: Continuity and Matrix Analysis. 284
Table of Contents Chapter 3: Tensors and Multilinearity The Kronecker Product. 3.1 xv 297 297 3.1.1 Definition and Basic Properties 298 3.1.2 Vectorization and the Swap Matrix 305 3.1.3 The Symmetric and Antisymmetric Subspaces 311 Exercises 318 3.2 Multilinear Transformations. 320 3.2.1 Definition and Basic Examples 320 3.2.2 Arrays 323 3.2.3 Properties of Multilinear Transformations 333 Exercises 340 The Tensor Product. 342 3.3 3.3.1 Motivation and Definition 342 3.3.2 Existence and Uniqueness 345 3.3.3 Tensor Rank 350 Exercises 356 3.4 Summary and Review. 357 3.A Extra Topic: Matrix-Valued Linear Maps. 358 3.B Extra Topic: Homogeneous Polynomials. 377 3.C Extra Topic: Semidefinite Programming. 393 Appendix A: Mathematical Preliminaries A.l Review of Introductory Linear Algebra. 415 415 A. 1.1 Systems of Linear Equations 415 A.1.2 Matrices as Linear Transformations 416 A. 1.3 The Inverse of a Matrix 417 A. 1.4 Range, Rank, Null Space, and Nullity 418 A. 1.5 Determinants and
Permutations 419 A. 1.6 Eigenvalues and Eigenvectors 421 A. 1.7 Diagonalization 422 A.2 Polynomials and Beyond. 424 A.2.1 Monomials, Binomials and Multinomials 424 A.2.2 Taylor Polynomials and Taylor Series 426
Table of Contents xvi A.3 Complex Numbers. 428 A.3.1 Basic Arithmetic and Geometry 429 A.3.2 The Complex Conjugate 430 A.3.3 Euler’s Formula and Polar Form 431 A.4 Fields. A.5 Convexity. 436 434 A.5.1 Convex Sets 436 A.5.2 Convex Functions 438 Appendix B: Additional Proofs 441 B.l Equivalence of Norms. B.2 Details of the Jordan Decomposition. 442 B.3 Strong Duality for Semidefinite Programs. 446 Appendix C: Selected Exercise Solutions 441 449 C.l Chapter 1 : Vector Spaces. 451 C.2 Chapter 2: Matrix Decompositions. 465 C.3 Chapter 3: Tensors and Multilinearity. 476 Bibliography. 487 Index. 489 Symbol
Index. 494 |
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| illustrated | Illustrated |
| index_date | 2024-07-03T20:26:39Z |
| indexdate | 2024-07-10T09:37:37Z |
| institution | BVB |
| isbn | 9783030528140 |
| language | English |
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| physical | xvi, 494 Seiten Illustrationen, Diagramme |
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| spelling | Johnston, Nathaniel (DE-588)1157532357 aut Advanced linear and matrix algebra Nathaniel Johnston Cham Springer [2021] © 2021 xvi, 494 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Linear and Multilinear Algebras, Matrix Theory Linear Algebra Matrix theory Algebra Algebras, Linear Algebra (DE-588)4001156-2 gnd rswk-swf Matrizentheorie (DE-588)4128970-5 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s Matrizentheorie (DE-588)4128970-5 s DE-604 Algebra (DE-588)4001156-2 s Erscheint auch als Online-Ausgabe 978-3-030-52815-7 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033794912&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Johnston, Nathaniel Advanced linear and matrix algebra Linear and Multilinear Algebras, Matrix Theory Linear Algebra Matrix theory Algebra Algebras, Linear Algebra (DE-588)4001156-2 gnd Matrizentheorie (DE-588)4128970-5 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4128970-5 (DE-588)4035811-2 |
| title | Advanced linear and matrix algebra |
| title_auth | Advanced linear and matrix algebra |
| title_exact_search | Advanced linear and matrix algebra |
| title_exact_search_txtP | Advanced linear and matrix algebra |
| title_full | Advanced linear and matrix algebra Nathaniel Johnston |
| title_fullStr | Advanced linear and matrix algebra Nathaniel Johnston |
| title_full_unstemmed | Advanced linear and matrix algebra Nathaniel Johnston |
| title_short | Advanced linear and matrix algebra |
| title_sort | advanced linear and matrix algebra |
| topic | Linear and Multilinear Algebras, Matrix Theory Linear Algebra Matrix theory Algebra Algebras, Linear Algebra (DE-588)4001156-2 gnd Matrizentheorie (DE-588)4128970-5 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Linear and Multilinear Algebras, Matrix Theory Linear Algebra Matrix theory Algebra Algebras, Linear Matrizentheorie Lineare Algebra |
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