Advanced Linear Algebra for Engineers with MATLAB

by Dianat; Sohail A.

• ISBN13: 9781420095234
• ISBN10: 1420095234
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-02-23
• Publisher: CRC Press

Summary

Designed to elevate the analytical and problem-solving skills of engineering students, this text provides systematic instruction that will allow those students to make full use of the advanced capacities that MATLAB Ò provides. Based on the applied experience of two leading industry consultants in signal and image processing and circuit analysis, this textbook is designed to support the highly regarded courses the two teach at RIT. Offering a broad selection of progressive exercises and MATLAB Ò problems, each chapter features carefully chosen examples that demonstrate underlying ideas at work in practical scenarios. A complete solutions manual is provided for qualifying instructors.

Table of Contents

Preface	p. xiii
Authors	p. xvii
Matrices, Matrix Algebra, and Elementary Matrix Operations	p. 1
Introduction	p. 1
Basic Concepts and Notation	p. 1
Matrix and Vector Notation	p. 1
Matrix Definition	p. 1
Elementary Matrices	p. 3
Elementary Matrix Operations	p. 5
Matrix Algebra	p. 6
Matrix Addition and Subtraction	p. 7
Properties of Matrix Addition	p. 7
Matrix Multiplication	p. 7
Properties of Matrix Multiplication	p. 8
Applications of Matrix Multiplication in Signal and Image Processing	p. 8
Application in Linear Discrete One Dimensional Convolution	p. 9
Application in Linear Discrete Two Dimensional Convolution	p. 14
Matrix Representation of Discrete Fourier Transform	p. 18
Elementary Row Operations	p. 22
Row Echelon Form	p. 23
Elementary Transformation Matrices	p. 24
Type 1: Scaling Transformation Matrix (E1	p. 24
Type 2: Interchange Transformation Matrix (E2)	p. 25
Type 3: Combination Transformation Matrices (E3)	p. 26
Solution of System of Linear Equations	p. 27
Gaussian Elimination	p. 27
Over Determined Systems	p. 31
Under Determined Systems	p. 32
Matrix Partitions	p. 32
Column Partitions	p. 33
Row Partitions	p. 34
Block Multiplication	p. 35
Inner, Outer, and Kronecker Products	p. 38
Inner Product	p. 38
Outer Product	p. 39
Kronecker Products	p. 40
Problems	p. 40
Determinants, Matrix Inversion and Solutions to Systems of Linear Equations	p. 49
Introduction	p. 49
Determinant of a Matrix	p. 49
Properties of Determinant	p. 52
Row Operations and Determinants	p. 53
Interchange of Two Rows	p. 53
Multiplying a Row of A by a Nonzero Constant	p. 54
Adding a Multiple of One Row to Another Row	p. 55
Singular Matrices	p. 55
Matrix Inversion	p. 58
Properties of Matrix Inversion	p. 60
Gauss-Jordan Method for Calculating Inverse of a Matrix	p. 60
Useful Formulas for Matrix Inversion	p. 63
Recursive Least Square (RLS) Parameter Estimation	p. 64
Solution of Simultaneous Linear Equations	p. 67
Equivalent Systems	p. 69
Strict Triangular Form	p. 69
Cramer's Rule	p. 70
LU Decomposition	p. 71
Applications: Circuit Analysis	p. 75
Homogeneous Coordinates System	p. 78
Applications of Homogeneous Coordinates in Image Processing	p. 79
Rank, Null Space and Invertibility of Matrices	p. 85
Null Space N(A)	p. 85
Column Space C(A)	p. 87
Row Space R(A)	p. 87
Rank of a Matrix	p. 89
Special Matrices with Applications	p. 90
Vandermonde Matrix	p. 90
Hankel Matrix	p. 91
Toeplitz Matrices	p. 91
Permutation Matrix	p. 92
Markov Matrices	p. 92
Circulant Matrices	p. 93
Hadamard Matrices	p. 93
Nilpotent Matrices	p. 94
Derivatives and Gradients	p. 95
Derivative of Scalar with Respect to a Vector	p. 95
Quadratic Functions	p. 96
Derivative of a Vector Function with Respect to a Vector	p. 98
Problems	p. 99
Linear Vector Spaces	p. 105
Introduction	p. 105
Linear Vector Space	p. 105
Definition of Linear Vector Space	p. 105
Examples of Linear Vector Spaces	p. 106
Additional Properties of Linear Vector Spaces	p. 107
Subspace of a Linear Vector Space	p. 107
Span of a Set of Vectors	p. 108
Spanning Set of a Vector Space	p. 110
Linear Dependence	p. 110
Basis Vectors	p. 113
Change of Basis Vectors	p. 114
Normed Vector Spaces	p. 116
Definition of Normed Vector Space	p. 116
Examples of Normed Vector Spaces	p. 116
Distance Function	p. 117
Equivalence of Norms	p. 118
Inner Product Spaces	p. 120
Definition of Inner Product	p. 120
Examples of Inner Product Spaces	p. 121
Schwarz's Inequality	p. 121
Norm Derived from Inner Product	p. 123
Applications of Schwarz Inequality in Communication Systems	p. 123
Detection of a Discrete Signal ôBuriedö in White Noise	p. 123
Detection of Continuous Signal ôBuriedö in Noise	p. 125
Hilbert Space	p. 129
Orthogonality	p. 131
Orthonormal Set	p. 131
Gram-Schmidt Orthogonalization Process	p. 131
Orthogonal Matrices	p. 134
Complete Orthonormal Set	p. 135
Generalized Fourier Series (GFS)	p. 135
Applications of GFS	p. 137
Continuous Fourier Series	p. 137
Discrete Fourier Transform (DFT)	p. 144
Legendre Polynomial	p. 145
Sinc Functions	p. 146
Matrix Factorization	p. 147
QR Factorization	p. 147
Solution of Linear Equations Using QR Factorization	p. 149
Problems	p. 151
Eigenvalues and Eigenvectors	p. 157
Introduction	p. 157
Matrices as Linear Transformations	p. 157
Definition: Linear Transformation	p. 157
Matrices as Linear Operators	p. 160
Null Space of a Matrix	p. 160
Projection Operator	p. 161
Orthogonal Projection	p. 162
Projection Theorem	p. 163
Matrix Representation of Projection Operator	p. 163
Eigenvalues and Eigenvectors	p. 165
Definition of Eigenvalues and Eigenvectors	p. 165
Properties of Eigenvalues and Eigenvectors	p. 168
Independent Property	p. 168
Product and Sum of Eigenvalues	p. 170
Finding the Characteristic Polynomial of a Matrix	p. 171
Modal Matrix	p. 173
Matrix Diagonalization	p. 173
Distinct Eigenvalues	p. 173
Jordan Canonical Form	p. 175
Special Matrices	p. 180
Unitary Matrices	p. 180
Hermitian Matrices	p. 183
Definite Matrices	p. 185
Positive Definite Matrices	p. 185
Positive Semidefinite Matrices	p. 185
Negative Definite Matrices	p. 185
Negative Semidefinite Matrices	p. 185
Test for Matrix Positiveness	p. 185
Singular Value Decomposition (SVD)	p. 188
Definition of SVD	p. 188
Matrix Norm	p. 192
Frobenius Norm	p. 195
Matrix Condition Number	p. 196
Numerical Computation of Eigenvalues and Eigenvectors	p. 199
Power Method	p. 199
Properties of Eigenvalues and Eigenvectors of Different Classes of Matrices	p. 205
Applications	p. 206
Image Edge Detection	p. 206
Gradient Based Edge Detection of Gray Scale Images	p. 209
Gradient Based Edge Detection of RGB Images	p. 210
Vibration Analysis	p. 214
Signal Subspace Decomposition	p. 217
Frequency Estimation	p. 217
Direction of Arrival Estimation	p. 219
Problems	p. 222
Matrix Polynomials and Functions of Square Matrices	p. 229
Introduction	p. 229
Matrix Polynomials	p. 229
Infinite Series of Matrices	p. 230
Convergence of an Infinite Matrix Series	p. 231
Cayley-Hamilton Theorem	p. 232
Matrix Polynomial Reduction	p. 234
Functions of Matrices	p. 236
Sylvester's Expansion	p. 236
Cayley-Hamilton Technique	p. 240
Modal Matrix Technique	p. 245
Special Matrix Functions	p. 246
Matrix Exponential Function eAt	p. 247
Matrix Function Ak	p. 249
The State Space Modeling of Linear Continuous-time Systems	p. 250
Concept of States	p. 250
State Equations of Continuous Time Systems	p. 250
State Space Representation of Continuous LTI Systems	p. 254
Solution of Continuous-time State Space Equations	p. 256
Solution of Homogenous State Equations and State Transition Matrix	p. 257
Properties of State Transition Matrix	p. 258
Computing State Transition Matrix	p. 258
Complete Solution of State Equations	p. 259
State Space Representation of Discrete-time Systems	p. 263
Definition of States	p. 263
State Equations	p. 263
State Space Representation of Discrete-time LTI Systems	p. 264
Solution of Discrete-time State Equations	p. 265
Solution of Homogenous State Equation and State Transition Matrix	p. 266
Properties of State Transition Matrix	p. 266
Computing the State Transition Matrix	p. 267
Complete Solution of the State Equations	p. 268
Controllability of LTI Systems	p. 270
Definition of Controllability	p. 270
Controllability Condition	p. 270
Observability of LTI Systems	p. 272
Definition of Observability	p. 272
Observability Condition	p. 272
Problems	p. 276
Introduction to Optimization	p. 283
Introduction	p. 283
Stationary Points of Functions of Several Variables	p. 283
Hessian Matrix	p. 285
Least-Square (LS) Technique	p. 287
LS Computation Using QR Factorization	p. 288
LS Computation Using Singtilar Value Decomposition (SVD)	p. 289
Weighted Least Square (WLS)	p. 291
LS Curve Fitting	p. 293
Applications of LS Technique	p. 295
One Dimensional Wiener Filter	p. 295
Choice of Q Matrix and Scale Factor ß	p. 298
Two Dimensional Wiener Filter	p. 300
Total Least-Squares (TLS)	p. 302
Eigen Filters	p. 304
Stationary Points with Equality Constraints	p. 307
Lagrange Multipliers	p. 307
Applications	p. 310
Maximum Entropy Problem	p. 310
Design of Digital Finite Impulse Response (FIR) Filters	p. 312
Problems	p. 316
The Laplace Transform	p. 321
Definition of the Laplace Transform	p. 321
The Inverse Laplace Transform	p. 323
Partial Fraction Expansion	p. 323
The z-Transform	p. 329
Definition of the z-Transform	p. 329
The Inverse z-Transform	p. 330
Inversion by Partial Fraction Expansion	p. 330
Bibliography	p. 335
Index	p. 339
Table of Contents provided by Ingram. All Rights Reserved.