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Contents
Preface ix
1 Matrices and Systems of Equations 1
1.1Systems of Linear Equations 1
1.2 RowEchelonForm11
1.3Matrix Arithmetic 27
1.4Matrix Algebra 46
1.5Elementary Matrices 60
1.6Partitioned Matrices 70
MATLABExercises80
ChapterTestA—TrueorFalse84
Chapter Test B 85
2 Determinants87
2.1The Determinant of a Matrix 87
2.2Properties of Determinants 94
2.3Additional Topics and Applications 101
MATLABExercises109
ChapterTestA—TrueorFalse111
Chapter Test B 111
3 Vector Spaces112
3.1 Definitioand Examples 112
3.2 Subspaces 119
3.3 Linear Independence 130
3.4 Basis and Dimensio141
3.5 Change of Basis 147
3.6 Row Space and ColumSpace 157
MATLABExercises165
ChapterTestA—TrueorFalse166
Chapter Test B 167
4 Linear Transformations169
4.1De.nitioand Examples 169
4.2Matrix Representations of Linear Transformations 178
4.3Similarity 192
MATLABExercises198
ChapterTestA—TrueorFalse199
Chapter Test B 200
5 Orthogonality201
5.1The Scalar Product iR202
5.2Orthogonal Subspaces 217
5.3Least Squares Problems 225
5.4Inner Product Spaces 238
5.5Orthonormal Sets 247
5.6The Gram–Schmidt OrthogonalizatioProcess 266
5.7Orthogonal Polynomials 275
MATLABExercises283
ChapterTestA—TrueorFalse285
Chapter Test B 285
6 Eigenvalues287
6.1Eigenvaluesand Eigenvectors288
6.2Systems of Linear Differential Equations 301
6.3Diagonalizatio312
6.4HermitiaMatrices 330
6.5The Singular Value Decompositio342
6.6Quadratic Forms 356
6.7Positive De.nite Matrices 370
6.8Nonnegative Matrices 377
MATLABExercises387
ChapterTestA—TrueorFalse393
Chapter Test B 393
7 Numerical Linear Algebra395
7.1Floating-Point Numbers 396
7.2GaussiaEliminatio404
7.3PivotingStrategies409
7.4Matrix Norms and ConditioNumbers 415
7.5Orthogonal Transformations 429
7.6The Eigenvalue Problem 440
7.7Least Squares Problems 451
MATLABExercises463
ChapterTestA—TrueorFalse468
Chapter Test B 468
8 Iterative MethodsOnline.
8.1Basic Iterative Methods
9 Canonical FormsOnline.
9.1Nilpotent Operators
9.2The JordaCanonical Form
Appendix:MATLAB471
Bibliography 483
Answers to Selected Exercises 486
Index 499