Linear algebra and matrix analysis for statistics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2014
|
| Series: | Texts in statistical science
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 565 S. |
| ISBN: | 9781420095388 |
Staff View
MARC
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| 100 | 1 | |a Banerjee, Sudipto |d 1972- |e Verfasser |0 (DE-588)134074467 |4 aut | |
| 245 | 1 | 0 | |a Linear algebra and matrix analysis for statistics |c Sudipto Banerjee ; Anindya Roy |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2014 | |
| 300 | |a XVII, 565 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Texts in statistical science | |
| 490 | 0 | |a A Chapman & Hall book | |
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| 653 | |a Algebras, Linear | ||
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Record in the Search Index
| _version_ | 1817733773190496256 |
|---|---|
| adam_text |
Contents
Preface
xv
1
Matrices, Vectors and Their Operations
1
1.1
Basic definitions and notations
2
1.2
Matrix addition and scalar-matrix multiplication
5
1.3
Matrix multiplication
7
1.4
Partitioned matrices
14
1.4.1 2
χ
2
partitioned matrices
14
1.4.2
General partitioned matrices
16
1.5
The "trace" of a square matrix
18
1.6
Some special matrices
20
1.6.1
Permutation matrices
20
1.6.2
Triangular matrices
22
1.6.3
Hessenberg
matrices
24
1.6.4
Sparse matrices
26
1.6.5
Banded matrices
27
1.7
Exercises
29
2
Systems of Linear Equations
33
2.1
Introduction
33
2.2
Gaussian elimination
34
2.3
Gauss-Jordan elimination
42
2.4
Elementary matrices
44
2.5
Homogeneous linear systems
48
2.6
The inverse of a matrix
51
2.7
Exercises
61
IX
CONTENTS
χ
3
More on Linear Equations
3.1
The
LU
decomposition 6^
3.2
Crout's Algorithm 71
3.3
LU
decomposition with row interchanges 72
3.4
The LOU and Cholesky factorizations
77
3.5
Inverse of partitioned matrices 79
3.6
The LOU decomposition for partitioned matrices
81
3.7
The Sherman-Woodbury-Morrison formula
81
3.8
Exercises H3
4
Euclidean Spaces
87
4.1
Introduction
87
4.2
Vector addition and scalar multiplication
87
4.3
Linear spaces and subspaces
89
4.4
Intersection and sum of subspaces
92
4.5
Linear combinations and spans
94
4.6
Four fundamental subspaces
97
4.7
Linear independence
103
4.8
Basis and dimension
116
4.9
Change of basis and similar matrices
123
4.10
Exercises
124
5
The Rank of a Matrix
129
5.1
Rank and nullity of a matrix
129
5.2
Bases for the four fundamental subspaces
1
38
5.3
Rank and inverse
141
5.4
Rank factorization 1
44
5.5
The rank-normal form
140
5.6
Rank of a partitioned matrix
j
5q
5.7
Bases for the fundamental subspaces using the rank normal form
151
5.8
Exercises 1
<.»
CONTENTS xi
6
Complementary
Subspaces 155
6.1
Sum of
subspaces 155
6.2
The dimension of the sum of subspaces
156
6.3
Direct sums and complements
159
6.4
Projectors
163
6.5
The column space-null space decomposition
170
6.6
Invariant subspaces and the Core-Nilpotent decomposition
172
6.7
Exercises
176
7
Orthogonality, Orthogonal Subspaces and Projections
179
7.1
Inner product, norms and orthogonality
179
7.2
Row rank
=
column rank: A proof using orthogonality
184
7.3
Orthogonal projections
185
7.4
Gram-Schmidt orthogonalization
191
7.5
Orthocomplementary subspaces
197
7.6
The Fundamental Theorem of Linear Algebra
200
7.7
Exercises
203
8
More on Orthogonality
207
8.1
Orthogonal matrices
207
8.2
The
Q R
decomposition
211
8.3
Orthogonal projection and projector
215
8.4
Orthogonal projector: Alternative derivations
220
8.5
Sum of orthogonal projectors
222
8.6
Orthogonal triangularization
225
8.6.1
The modified Gram-Schmidt process
226
8.6.2
Reflectors
227
8.6.3
Rotations
232
8.6.4
The rectangular QR decomposition
239
8.6.5
Computational effort
240
8.7
Orthogonal similarity reduction to
Hessenberg
forms
243
8.8
Orthogonal reduction to
bidiagonal
forms
248
8.9
Some further reading on statistical linear models
252
8.10
Exercises
253
CONTENTS
xii
9
Revisiting Linear Equations
, . 257
9.1
Introduction
9.2
Null spaces and the general solution of linear systems
257
259
9.3
Rank and linear systems
9.4
Generalized inverse of a matrix
9.5
Generalized inverses and linear systems
9.6
The Moore-Penrose inverse
974
9.7
Exercises
^'
10
Determinants
277
10.1
Introduction 277
10.2
Some basic properties of determinants
280
10.3
Determinant of products
288
10.4
Computing determinants
290
10.5
The determinant of the transpose of a matrix
—
revisited
291
10.6
Determinants of partitioned matrices
292
10.7
Cofactors and expansion theorems
297
10.8
The minor and the rank of a matrix
300
10.9
The Cauchy-Binet formula
303
10.10
The
Lapiace
expansion
306
10.11
Exercises
308
11
Eigenvalues and Eigenvectors
311
11.1
The Eigenvalue equation
3
13
11.2
Characteristic polynomial and its roots
316
11.3 Eigenspaces
and multiplicities
320
il A
Diagonalizable matrices
325
11.5
Similarity with triangular matrices
334
11.6
Matrix polynomials and the Caley-Hamilton Theorem
338
11.7
Spectral decomposition of real symmetric matrices
348
11.8
Computation of eigenvalues
352
11.9
Exercises
CONTENTS xiii
12 Singular
Value and
Jordan
Decompositions
371
12.1 Singular
value decomposition
371
12.2
The
SVD
and the four fundamental subspaces
379
12.3
SVD
and linear systems
381
12.4
SVD,
data compression and principal components
383
12.5
Computing the
SVD
385
12.6
The Jordan Canonical Form
389
12.7
Implications of the Jordan Canonical Form
397
12.8
Exercises
399
13
Quadratic Forms
401
13.1
Introduction
401
13.2
Quadratic forms
402
13.3
Matrices in quadratic forms
405
13.4
Positive and
nonnegative
definite matrices
411
13.5
Congruence and Sylvester's Law of Inertia
419
13.6
Nonnegative
definite matrices and minors
423
13.7
Some inequalities related to quadratic forms
425
13.8
Simultaneous diagonalization and the generalized eigenvalue prob¬
lem
434
13.9
Exercises
441
14
The
Kronecker
Product and Related Operations
445
14.1
Bilinear interpolation and the
Kronecker
product
445
14.2
Basic properties of
Kronecker
products
446
14.3
Inverses, rank and nonsingularity of
Kronecker
products
453
14.4
Matrix factorizations for
Kronecker
products
455
14.5
Eigenvalues and determinant
460
14.6
The
vec
and commutator operators
461
14.7
Linear systems involving
Kronecker
products
466
14.8
Sylvester's equation and the
Kronecker
sum
470
14.9
The
Hadamard
product
472
14.10
Exercises
480
CONTENTS
XIV
15 Linear Iterative Systems,
Norms and Convergence
483
15.1 Linear iterative
systems and convergence of matrix powers
483
JOC
15.2
Vector norms
^0^
15.3
Spectral radius and matrix convergence
489
15.4
Matrix norms and the Gerschgorin circles
491
15.5
The singular value decomposition
—
revisited
499
15.6
Web page ranking and Markov chains
503
15.7
Iterative algorithms for solving linear equations
51 1
15.7.1
The Jacobi method
512
15.7.2
The Gauss-Seidel method
513
15.7.3
The Successive Over-Relaxation
(SOR)
method
514
15.7.4
The conjugate gradient method
5
1
4
15.8
Exercises
517
16
Abstract Linear Algebra
521
16.1
General vector spaces
521
16.2
General inner products
528
16.3
Linear transformations, adjoint and rank
531
16.4
The four fundamental subspaces
—
revisited
535
16.5
Inverses of linear transformations
537
16.6
Linear transformations and matrices
54O
16.7
Change of bases, equivalence and similar matrices
543
16.8
Hubert spaces
547
16.9
Exercises
References
Index
559 |
| any_adam_object | 1 |
| author | Banerjee, Sudipto 1972- Roy, Anindya 1970- |
| author_GND | (DE-588)134074467 (DE-588)105254603X |
| author_facet | Banerjee, Sudipto 1972- Roy, Anindya 1970- |
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| ctrlnum | (OCoLC)882092888 (DE-599)BSZ352989106 |
| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| indexdate | 2024-12-06T23:00:12Z |
| institution | BVB |
| isbn | 9781420095388 |
| language | English |
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| series2 | Texts in statistical science A Chapman & Hall book |
| spelling | Banerjee, Sudipto 1972- Verfasser (DE-588)134074467 aut Linear algebra and matrix analysis for statistics Sudipto Banerjee ; Anindya Roy Boca Raton [u.a.] CRC Press 2014 XVII, 565 S. txt rdacontent n rdamedia nc rdacarrier Texts in statistical science A Chapman & Hall book Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Linear models (Statistics) Algebras, Linear Matrices Vector analysis Lineare Algebra (DE-588)4035811-2 s DE-604 Roy, Anindya 1970- Verfasser (DE-588)105254603X aut Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026999261&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Banerjee, Sudipto 1972- Roy, Anindya 1970- Linear algebra and matrix analysis for statistics Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 |
| title | Linear algebra and matrix analysis for statistics |
| title_auth | Linear algebra and matrix analysis for statistics |
| title_exact_search | Linear algebra and matrix analysis for statistics |
| title_full | Linear algebra and matrix analysis for statistics Sudipto Banerjee ; Anindya Roy |
| title_fullStr | Linear algebra and matrix analysis for statistics Sudipto Banerjee ; Anindya Roy |
| title_full_unstemmed | Linear algebra and matrix analysis for statistics Sudipto Banerjee ; Anindya Roy |
| title_short | Linear algebra and matrix analysis for statistics |
| title_sort | linear algebra and matrix analysis for statistics |
| topic | Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026999261&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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