Matrix operations for engineers and scientists: an essential guide in linear algebra
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Springer
2010
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XI, 314 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9789048192731 |
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MARC
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Record in the Search Index
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| adam_text |
CONTENTS
1 MATRICES AND LINEAR SYSTEMS OF EQUATIONS
. 1
1.1 SYSTEMS OF ALGEBRAIC EQUATIONS
. 1
1.2 SUFFIX AND MATRIX NOTATION
. 3
1.3 EQUALITY, ADDITION AND SCALING OF MATRICES . .
. 4
1.4 SOME SPECIAL MATRICES AND THE TRANSPOSE OPERATION .
6
2 DETERMINANTS, AND LINEAR INDEPENDENCE
. 13
2.1 INTRODUCTION TO DETERMINANTS AND SYSTEMS OF EQUATIONS .
13
2.2 A FIRST LOOK AT LINEAR DEPENDENCE AND INDEPENDENCE . 15
2.3 PROPERTIES OF DETERMINANTS AND THE LAPLACE EXPANSION THEOREM . . .
16
2.4 GAUSSIAN ELIMINATION AND DETERMINANTS .
25
2.5 HOMOGENEOUS SYSTEMS OF EQUATIONS AND A TEST FOR LINEAR
INDEPENDENCE .
28
2.6 DETERMINANTS AND EIGENVALUES: A FIRST LOOK .
30
3 MATRIX MULTIPLICATION, THE INVERSE MATRIX AND PARTITIONING
. 35
3.1 THE INNER PRODUCT, ORTHOGONALITY AND THE NORM .
35
3.1.1 A DIGRESSION ON NORMS . 36
3.2 MATRIX MULTIPLICATION
. 37
3.3 QUADRATIC FORMS
. 42
3.4 THE INVERSE MATRIX
. 45
3.5 ORTHOGONAL MATRICES
. 50
3.6 A MATRIX PROOF OF CRAMER'S RULE .
52
3.7 PARTITIONING OF MATRICES
. 54
3.8 MATRICES AND LEAST-SQUARES CURVE FITTING . . .
. 59
3.9 MATRICES AND THE LAPLACE EQUATION
. 64
4 SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS
. 75
4.1 THE AUGMENTED MATRIX AND ELEMENTARY ROW OPERATIONS . 75
4.2 THE ECHELON AND REDUCED ECHELON FORMS OF A MATRIX . 78
IX
4.3 THE ROW RANK OF A MATRIX .
80
4.3.1 ROW RANK OF AN AUGMENTED MATRIX AND THE NATURE
OF A SOLUTION SET . 84
4.3.2 TESTING THE LINEAR INDEPENDENCE OF THE ROWS (COLUMNS)
OF AN
N
N
MATRIX A . 85
4.4 ELEMENTARY ROW OPERATIONS AND THE INVERSE MATRIX . 85
4.5 LU FACTORIZATION OF A MATRIX AND ITS USE WHEN SOLVING LINEAR
SYSTEMS OF ALGEBRAIC EQUATIONS . 86
4.6 EIGENVALUES AND EIGENVECTORS
. 91
4.7 THE COMPANION MATRIX AND THE CHARACTERISTIC POLYNOMIAL . 95
5 EIGENVALUES, EIGENVECTORS, DIAGONALIZATION, SIMILARITY, JORDAN
NORMAL FORMS, AND ESTIMATING REGIONS CONTAINING EIGENVALUES
. . 101
5.1 FINDING EIGENVECTORS .
101
5.1.1 THE ROOTS OF POLYNOMIALS WITH REAL COEFFICIENTS . 106
5.2 DIAGONALIZATION OF MATRICES .
110
5.2.1 PROPERTIES OF DIAGONALIZED MATRICES . . .
112
5.3 QUADRATIC FORMS AND DIAGONALIZATION .
114
5.3.1 THE CLASSIFICATION OF QUADRATIC FORMS . 116
5.4 THE CHARACTERISTIC POLYNOMIAL AND THE CAYLEY*HAMILTON
THEOREM .
121
5.5 EIGENVALUES AND THE TRANSPOSE OPERATION . . . .
122
5.6 SIMILAR MATRICES
. 122
5.6.1 SIMILAR MATRICES .
123
5.7 LEFT AND RIGHT EIGENVECTORS .
124
5.8 JORDAN NORMAL FORMS .
125
5.9 A SPECIAL TRIDIAGONAL MATRIX, ITS EIGENVALUES AND
EIGENVECTORS .
133
5.10 THE POWER METHOD FOR EIGENVALUES AND EIGENVECTORS . 137
5.11 ESTIMATING REGIONS CONTAINING EIGENVALUES . 141
5.12 THE FIBONACCI SEQUENCE AND MATRICES .
145
5.13 A TWO-POINT BOUNDARY-VALUE PROBLEM AND A
TRIDIAGONAL MATRIX . 146
5.14 MATRICES WITH COMPLEX ELEMENTS .
148
5.14.1 HERMITIAN MATRICES . 148
5.14.2 SKEW-HERMITIAN MATRICES . 151
5.14.3 UNITARY MATRICES . 151
6 SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
. 159
6.1 DIFFERENTIATION AND INTEGRATION OF MATRICES . .
. 159
6.2 SYSTEMS OF HOMOGENEOUS CONSTANT COEFFICIENT DIFFERENTIAL
EQUATIONS .
163
6.2.1 THE HOMOGENEOUS SYSTEM . 164
6.3 AN APPLICATION OF DIAGONALIZATION .
169
X CONTENTS
6.4 THE NONHOMOGENEOUS CASE . 172
6.5 MATRIX METHODS AND THE LAPLACE TRANSFORM . .
175
6.6 THE MATRIX EXPONENTIAL AND DIFFERENTIAL EQUATIONS .
182
6.6.1 FINDING THE MATRIX EXPONENTIAL USING
THE LAPLACE TRANSFORM . 197
APPENDIX 1: THE SOLUTION OF A LINEAR FIRST-ORDER
DIFFERENTIAL EQUATION .
198
APPENDIX 2: A SUMMARY OF THE LAPLACE TRANSFORM AND A SHORT
TABLE OF LAPLACE TRANSFORM PAIRS . 199
7 AN INTRODUCTION TO VECTOR SPACES
. 207
7.1 A GENERALIZATION OF VECTORS .
207
7.2 VECTOR SPACES AND A BASIS FOR A VECTOR SPACE .
211
7.3 CHANGING BASIS VECTORS .
216
7.4 ROW AND COLUMN RANK .
216
7.5 THE INNER PRODUCT
. 218
7.6 THE ANGLE BETWEEN VECTORS AND ORTHOGONAL PROJECTIONS . 222
7.7 GRAM*SCHMIDT ORTHOGONALIZATION .
224
7.7.1 THE GRAM*SCHMIDT ORTHOGONALIZATION PROCESS IN
R
3
. 224
7.7.2 THE EXTENSION OF THE GRAM*SCHMIDT ORTHOGONALIZATION
PROCESS TO
R
N
. 225
7.8 PROJECTIONS
. 227
7.9 SOME COMMENTS ON INFINITE-DIMENSIONAL VECTOR SPACES . 231
8 LINEAR TRANSFORMATIONS AND THE GEOMETRY OF THE PLANE
. 239
8.1 ROTATION OF COORDINATE AXES .
239
8.2 LINEAR TRANSFORMATIONS .
245
8.3 THE NULL SPACE OF A LINEAR TRANSFORMATION AND ITS RANGE . 255
8.4 LINEAR TRANSFORMATIONS AND THE GEOMETRY OF THE PLANE . 257
8.4.1 ROTATION ABOUT THE ORIGIN . 262
8.4.2 A REFLECTION IN A LINE
L
THROUGH THE ORIGIN . 262
8.4.3 THE ORTHOGONAL PROJECTION OF A POINT
P
ONTO A LINE
L
THROUGH THE ORIGIN . 263
8.4.4 SCALING IN THE
X
1
AND
X
2
DIRECTIONS . 264
8.4.5 A SHEAR .
265
8.4.6 COMPOSITE TRANSFORMATIONS . 266
8.4.7 THE TRANSFORMATION OF CURVES . 267
SOLUTIONS FOR ALL EXERCISES
. 273
INDEX
.
309
CONTENTS XI |
| any_adam_object | 1 |
| author | Jeffrey, Alan 1929-2010 |
| author_GND | (DE-588)13561208X |
| author_facet | Jeffrey, Alan 1929-2010 |
| author_role | aut |
| author_sort | Jeffrey, Alan 1929-2010 |
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| bvnumber | BV037196197 |
| classification_rvk | SK 220 |
| ctrlnum | (OCoLC)700371012 (DE-599)DNB1001079698 |
| discipline | Physik Mathematik |
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| indexdate | 2024-07-20T10:57:04Z |
| institution | BVB |
| isbn | 9789048192731 |
| language | English |
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| physical | XI, 314 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2010 |
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| spelling | Jeffrey, Alan 1929-2010 Verfasser (DE-588)13561208X aut Matrix operations for engineers and scientists an essential guide in linear algebra Alan Jeffrey Dordrecht [u.a.] Springer 2010 XI, 314 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lineares Differentialgleichungssystem (DE-588)4452554-0 gnd rswk-swf Matrizenrechnung (DE-588)4126963-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Matrizenrechnung (DE-588)4126963-9 s Lineares Differentialgleichungssystem (DE-588)4452554-0 s DE-604 Erscheint auch als Online-Ausgabe 978-90-481-9274-8 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3447245&prov=M&dok_var=1&dok_ext=htm Inhaltstext SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021110540&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jeffrey, Alan 1929-2010 Matrix operations for engineers and scientists an essential guide in linear algebra Lineares Differentialgleichungssystem (DE-588)4452554-0 gnd Matrizenrechnung (DE-588)4126963-9 gnd |
| subject_GND | (DE-588)4452554-0 (DE-588)4126963-9 (DE-588)4123623-3 |
| title | Matrix operations for engineers and scientists an essential guide in linear algebra |
| title_auth | Matrix operations for engineers and scientists an essential guide in linear algebra |
| title_exact_search | Matrix operations for engineers and scientists an essential guide in linear algebra |
| title_full | Matrix operations for engineers and scientists an essential guide in linear algebra Alan Jeffrey |
| title_fullStr | Matrix operations for engineers and scientists an essential guide in linear algebra Alan Jeffrey |
| title_full_unstemmed | Matrix operations for engineers and scientists an essential guide in linear algebra Alan Jeffrey |
| title_short | Matrix operations for engineers and scientists |
| title_sort | matrix operations for engineers and scientists an essential guide in linear algebra |
| title_sub | an essential guide in linear algebra |
| topic | Lineares Differentialgleichungssystem (DE-588)4452554-0 gnd Matrizenrechnung (DE-588)4126963-9 gnd |
| topic_facet | Lineares Differentialgleichungssystem Matrizenrechnung Lehrbuch |
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