A polynomial approach to linear algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2012
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 411 S. graph. Darst. |
| ISBN: | 9781461403371 |
Internformat
MARC
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| 100 | 1 | |a Fuhrmann, Paul Abraham |d 1937- |e Verfasser |0 (DE-588)141517077 |4 aut | |
| 245 | 1 | 0 | |a A polynomial approach to linear algebra |c Paul A. Fuhrmann |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2012 | |
| 300 | |a XVI, 411 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Universitext | |
| 650 | 4 | |a Lineare Algebra - Lehrbuch | |
| 650 | 4 | |a Algebras, Linear | |
| 650 | 0 | 7 | |a Lineare Algebra |0 (DE-588)4035811-2 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Lineare Algebra |0 (DE-588)4035811-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4614-0338-8 |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-024815470 | ||
Datensatz im Suchindex
| _version_ | 1804148931406331904 |
|---|---|
| adam_text | Contents
Algebraic Preliminaries
.................................................... 1
1.1
Introduction
........................................................... 1
1.2
Sets and Maps
........................................................ 1
1.3
Groups
................................................................ 3
1.4
Rings and Fields
...................................................... 8
1.4.1
The Integers
................................................. 11
1.4.2
The Polynomial Ring
....................................... 11
1.4.3
Formal Power Series
........................................ 20
1.4.4
Rational Functions
.......................................... 22
1.4.5
Proper Rational Functions
.................................. 23
1.4.6
Stable Rational Functions
.................................. 24
1.4.7
Truncated Laurent Series
................................... 25
1.5
Modules
............................................................... 26
1.6
Exercises
.............................................................. 31
1.7
Notes and Remarks
................................................... 32
Vector Spaces
................................................................ 33
2.1
Introduction
........................................................... 33
2.2
Vector Spaces
......................................................... 33
2.3
Linear Combinations
................................................. 36
2.4
Subspaces
............................................................. 36
2.5
Linear Dependence and Independence
.............................. 37
2.6
Subspaces and Bases
................................................. 40
2.7
Direct Sums
........................................................... 41
2.8
Quotient Spaces
...................................................... 43
2.9
Coordinates
........................................................... 45
2.10
Change of Basis Transformations
................................... 46
2.11 Lagrange
Interpolation
............................................... 48
2.12
Taylor Expansion
..................................................... 51
2.13
Exercises
.............................................................. 52
2.14
Notes and Remarks
................................................... 52
xiv
Contents
3
Determinants
................................................................ 55
3.1
Introduction
........................................................... 55
3.2
Basic Properties
...................................................... 55
3.3
Cramer s Rule
........................................................ 60
3.4
The Sylvester Resultant
.............................................. 62
3.5
Exercises
.............................................................. 64
3.6
Notes and Remarks
................................................... 66
4
Linear Transformations
.................................................... 67
4.1
Introduction
........................................................... 67
4.2
Linear Transformations
.............................................. 67
4.3
Matrix Representations
.............................................. 75
4.4
Linear Functionals and Duality
...................................... 79
4.5
The Adjoint Transformation
......................................... 85
4.6
Polynomial Module Structure on Vector Spaces
.................... 88
4.7
Exercises
.............................................................. 94
4.8
Notes and Remarks
................................................... 95
5
The Shift Operator
......................................................... 97
5.1
Introduction
........................................................... 97
5.2
Basic Properties
...................................................... 97
5.3
Circulant
Matrices
.................................................... 109
5.4
Rational Models
......................................................
Ill
5.5
The Chinese Remainder Theorem and Interpolation
............... 118
5.5.1 Lagrange
Interpolation Revisited
.......................... 119
5.5.2
Hermite Interpolation
....................................... 120
5.5.3
Newton Interpolation
....................................... 121
5.6
Duality
................................................................ 122
5.7
Universality of Shifts
................................................. 127
5.8
Exercises
.............................................................. 131
5.9
Notes and Remarks
................................................... 133
6
Structure Theory of Linear Transformations
........................... 135
6.1
Introduction
........................................................... 135
6.2
Cyclic Transformations
.............................................. 135
6.2.1
Canonical Forms for Cyclic Transformations
............. 140
6.3
The Invariant Factor Algorithm
...................................... 145
6.4
Noncyclic Transformations
.......................................... 147
6.5
Diagonalization
....................................................... 151
6.6
Exercises
.............................................................. 154
6.7
Notes and Remarks
................................................... 158
7
Inner Product Spaces
...................................................... 161
7.1
Introduction
........................................................... 161
7.2
Geometry of Inner Product Spaces
.................................. 161
7.3
Operators in Inner Product Spaces
.................................. 166
7.3.1
The Adjoint Transformation
............................... 166
Contents xv
7.3.2
Unitary
Operators........................................... 169
7.3.3
Self-adjoint
Operators...................................... 173
7.3.4 The Minimax
Principle
..................................... 176
7.3.5 The Cayley Transform...................................... 176
7.3.6 Normal Operators........................................... 178
7.3.7 Positive Operators.......................................... 180
7.3.8
Partial
Isometries........................................... 182
7.3.9 The Polar
Decomposition
.................................. 183
7.4 Singular
Vectors and
Singular
Values
............................... 184
7.5
Unitary Embeddings
................................................. 187
7.6
Exercises..............................................................
190
7.7 Notes
and Remarks
................................................... 193
8 Tensor Products
and Forms
............................................... 195
8.1
Introduction...........................................................
195
8.2 Basics................................................................. 196
8.2.1
Forms in
Inner
Product
Spaces............................. 196
8.2.2
Sylvester s Law of Inertia
.................................. 199
8.3
Some Classes of Forms
.............................................. 204
8.3.1
Hankel Forms
............................................... 205
8.3.2
Bezoutians
.................................................. 209
8.3.3
Representation of Bezoutians
.............................. 213
8.3.4
Diagonalization of Bezoutians
............................. 217
8.3.5
Bezout and Hankel Matrices
............................... 223
8.3.6
Inversion of Hankel Matrices
.............................. 230
8.3.7
Continued Fractions and Orthogonal Polynomials
........ 237
8.3.8
The Cauchy Index
.......................................... 248
8.4
Tensor Products of Models
.......................................... 254
8.4.1
Bilinear Forms
.............................................. 254
8.4.2
Tensor Products of Vector Spaces
......................... 255
8.4.3
Tensor Products of Modules
............................... 259
8.4.4 Kronecker
Product Models
................................. 260
8.4.5
Tensor Products over a Field
............................... 261
8.4.6
Tensor Products over the Ring of Polynomials
............ 263
8.4.7
The Polynomial Sylvester Equation
....................... 266
8.4.8
Reproducing Kernels
....................................... 270
8.4.9
The Bezout Map
............................................ 272
8.5
Exercises
.............................................................. 274
8.6
Notes and Remarks
................................................... 277
9
Stability
...................................................................... 279
9.1
Introduction
........................................................... 279
9.2
Root Location Using Quadratic Forms
.............................. 279
9.3
Exercises
.............................................................. 293
9.4
Notes and Remarks
................................................... 294
xvi Contents
10 Elements
of
Linear System
Theory
....................................... 295
10.1
Introduction...........................................................
295
10.2 Systems
and Their Representations.................................
296
10.3
Realization Theory
................................................... 300
10.4
Stabilization
.......................................................... 314
10.5
The Youla-Kucera Parametrization
................................. 319
10.6
Exercises
.............................................................. 321
10.7
Notes and Remarks
................................................... 324
11
Rational Hardy Spaces
..................................................... 325
11.1
Introduction
........................................................... 325
11.2
Hardy Spaces and Their Maps
....................................... 326
11.2.1
Rational Hardy Spaces
..................................... 326
11.2.2
Invariant Subspaces
......................................... 334
11.2.3
Model Operators and Intertwining Maps
.................. 337
11.2.4
Intertwining Maps and Interpolation
...................... 344
11.2.5
RHŢ-Chinese
Remainder Theorem
....................... 353
11.2.6
Analytic Hankel Operators and Intertwining Maps
....... 354
11.3
Exercises
.............................................................. 358
11.4
Notes and Remarks
................................................... 359
12
Model Reduction
............................................................ 361
12.1
Introduction
........................................................... 361
12.2
Hankel Norm Approximation
........................................ 362
12.2.1
Schmidt Pairs of Hankel Operators
........................ 363
12.2.2
Reduction to Eigenvalue Equation
......................... 369
12.2.3
Zeros of Singular Vectors and a Bezout equation
......... 370
12.2.4
More on Zeros of Singular Vectors
........................ 376
12.2.5
Nehari s Theorem
........................................... 378
12.2.6
Nevanlinna-Pick Interpolation
............................. 379
12.2.7
Hankel Approximant Singular Values and Vectors
....... 381
12.2.8
Orthogonality Relations
.................................... 383
12.2.9
Duality in Hankel Norm Approximation
.................. 385
12.3
Model Reduction: A Circle of Ideas
................................. 392
12.3.1
The Sylvester Equation and Interpolation
................. 392
12.3.2
The Sylvester Equation and the Projection Method
....... 394
12.4
Exercises
.............................................................. 397
12.5
Notes and Remarks
................................................... 400
References
......................................................................... 403
Index
............................................................................... 407
|
| any_adam_object | 1 |
| author | Fuhrmann, Paul Abraham 1937- |
| author_GND | (DE-588)141517077 |
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| callnumber-search | QA184.F8 1996 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 220 |
| classification_tum | MAT 150f |
| ctrlnum | (OCoLC)774398339 (DE-599)BVBBV039957673 |
| dewey-full | 512.5 512/.520 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.5 512/.5 20 |
| dewey-search | 512.5 512/.5 20 |
| dewey-sort | 3512.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV039957673 |
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| indexdate | 2024-07-10T00:14:58Z |
| institution | BVB |
| isbn | 9781461403371 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024815470 |
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| physical | XVI, 411 S. graph. Darst. |
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| spelling | Fuhrmann, Paul Abraham 1937- Verfasser (DE-588)141517077 aut A polynomial approach to linear algebra Paul A. Fuhrmann 2. ed. New York [u.a.] Springer 2012 XVI, 411 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Lineare Algebra - Lehrbuch Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Lineare Algebra (DE-588)4035811-2 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4614-0338-8 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024815470&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fuhrmann, Paul Abraham 1937- A polynomial approach to linear algebra Lineare Algebra - Lehrbuch Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 (DE-588)4123623-3 |
| title | A polynomial approach to linear algebra |
| title_auth | A polynomial approach to linear algebra |
| title_exact_search | A polynomial approach to linear algebra |
| title_full | A polynomial approach to linear algebra Paul A. Fuhrmann |
| title_fullStr | A polynomial approach to linear algebra Paul A. Fuhrmann |
| title_full_unstemmed | A polynomial approach to linear algebra Paul A. Fuhrmann |
| title_short | A polynomial approach to linear algebra |
| title_sort | a polynomial approach to linear algebra |
| topic | Lineare Algebra - Lehrbuch Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Lineare Algebra - Lehrbuch Algebras, Linear Lineare Algebra Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024815470&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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