ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.

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1 ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business

2 Contents Preface Authors xiii xvii 1 Matrices, Matrix Algebra, and Elementary Matrix Operations Introduction Basic Concepts and Notation Matrix and Vector Notation Matrix Definition Elementary Matrices Elementary Matrix Operations Matrix Algebra Matrix Addition and Subtraction Properties of Matrix Addition Matrix Multiplication Properties of Matrix Multiplication Applications of Matrix Multiplication in Signal and Image Processing Application in Linear Discrete One Dimensional Convolution Application in Linear Discrete Two Dimensional Convolution Matrix Representation of Discrete Fourier Transform Elementary Row Operations Row Echelon Form Elementary Transformation Matrices Type 1: Scaling Transformation Matrix (Ej) Type 2: Interchange Transformation Matrix (E 2 ) Type 3: Combination Transformation Matrices (E 3 ) Solution of System of Linear Equations Gaussian Elimination Over Determined Systems Under Determined Systems Matrix Partitions Column Partitions Row Partitions Block Multiplication 35 vii

3 viii Contents 1.8 Inner, Outer, and Kronecker Products Inner Product Outer Product Kronecker Products 40 Problems 40 2 Determinants, Matrix Inversion and Solutions to Systems of Linear Equations Introduction Determinant of a Matrix Properties of Determinant Row Operations and Determinants Interchange of Two Rows Multiplying a Row of A by a Nonzero Constant Adding a Multiple of One Row to Another Row Singular Matrices Matrix Inversion Properties of Matrix Inversion Gauss-Jordan Method for Calculating Inverse of a Matrix Useful Formulas for Matrix Inversion Recursive Least Square (RLS) Parameter Estimation Solution of Simultaneous Linear Equations Equivalent Systems Strict Triangular Form Cramer's Rule LU Decomposition Applications: Circuit Analysis Homogeneous Coordinates System Applications of Homogeneous Coordinates in Image Processing Rank, Null Space and Invertibility of Matrices Null Space MA) Column Space C(A) Row Space R(A) Rank of a Matrix Special Matrices with Applications Vandermonde Matrix Hankel Matrix Toeplitz Matrices Permutation Matrix Markov Matrices Circulant Matrices Hadamard Matrices Nilpotent Matrices 94

4 Contents ix 2.9 Derivatives and Gradients Derivative of Scalar with Respect to a Vector Quadratic Functions Derivative of a Vector Function with Respect to a Vector 98 Problems 99 3 Linear Vector Spaces Introduction Linear Vector Space Definition of Linear Vector Space Examples of Linear Vector Spaces Additional Properties of Linear Vector Spaces Subspace of a Linear Vector Space Span of a Set of Vectors Spanning Set of a Vector Space Linear Dependence Basis Vectors Change of Basis Vectors Normed Vector Spaces Definition of Normed Vector Space Examples of Normed Vector Spaces Distance Function Equivalence of Norms Inner Product Spaces Definition of Inner Product Examples of Inner Product Spaces Schwarz's Inequality Norm Derived from Inner Product Applications of Schwarz Inequality in Communication Systems Detection of a Discrete Signal "Buried" in White Noise Detection of Continuous Signal "Buried" in Noise Hilbert Space Orthogonality Orthonormal Set Gram-Schmidt Orthogonalization Process Orthogonal Matrices Complete Orthonormal Set Generalized Fourier Series (GFS) Applications of GFS Continuous Fourier Series Discrete Fourier Transform (DFT) Legendre Polynomial Sine Functions 146

5 x Contents 3.7 Matrix Factorization QR Factorization Solution of Linear Equations Using QR Factorization 149 Problems Eigenvalues and Eigenvectors Introduction ' Matrices as Linear Transformations Definition: Linear Transformation Matrices as Linear Operators Null Space of a Matrix Projection Operator Orthogonal Projection Projection Theorem Matrix Representation of Projection Operator Eigenvalues and Eigenvectors Definition of Eigenvalues and Eigenvectors Properties of Eigenvalues and Eigenvectors Independent Property Product and Sum of Eigenvalues Finding the Characteristic Polynomial of a Matrix Modal Matrix Matrix Diagonalization Distinct Eigenvalues Jordan Canonical Form Special Matrices Unitary Matrices Hermitian Matrices Definite Matrices Positive Definite Matrices Positive Semidefinite Matrices Negative Definite Matrices Negative Semidefinite Matrices Test for Matrix Positiveness Singular Value Decomposition (SVD) Definition of SVD Matrix Norm Frobenius Norm Matrix Condition Number Numerical Computation of Eigenvalues and Eigenvectors Power Method Properties of Eigenvalues and Eigenvectors of Different Classes of Matrices Applications Image Edge Detection 206

6 Contents xi Gradient Based Edge Detection of Gray Scale Images Gradient Based Edge Detection of RGB Images Vibration Analysis Signal Subspace Decomposition Frequency Estimation Direction of Arrival Estimation 219 Problems Matrix Polynomials and Functions of Square Matrices Introduction Matrix Polynomials Infinite Series of Matrices Convergence of an Infinite Matrix Series Cayley-Hamilton Theorem Matrix Polynomial Reduction Functions of Matrices Sylvester's Expansion Cayley-Hamilton Technique Modal Matrix Technique Special Matrix Functions Matrix Exponential Function e At Matrix Function A* The State Space Modeling of Linear Continuous-time Systems Concept of States State Equations of Continuous Time Systems State Space Representation of Continuous LTI Systems Solution of Continuous-time State Space Equations Solution of Homogenous State Equations and State Transition Matrix Properties of State Transition Matrix Computing State Transition Matrix Complete Solution of State Equations State Space Representation of Discrete-time Systems Definition of States State Equations State Space Representation of Discrete-time LTI Systems Solution of Discrete-time State Equations Solution of Homogenous State Equation and State Transition Matrix Properties of State Transition Matrix 266

7 xii Contents Computing the State Transition Matrix Complete Solution of the State Equations Controllability of LTI Systems Definition of Controllability Controllability Condition Observability of LTI Systems Definition of Observability Observability Condition 272 Problems Introduction to Optimization Introduction Stationary Points of Functions of Several Variables Hessian Matrix Least-Square (LS) Technique LS Computation Using QR Factorization LS Computation Using Singular Value Decomposition (SVD) Weighted Least Square (WLS) LS Curve Fitting Applications of LS Technique One Dimensional Wiener Filter Choice of Q Matrix and Scale Factor ß Two Dimensional Wiener Filter Total Least-Squares (TLS) Eigen Filters Stationary Points with Equality Constraints Lagrange Multipliers Applications Maximum Entropy Problem Design of Digital Finite Impulse Response (FIR) Filters 312 Problems 316 Appendix A: The Laplace Transform 321 Al Definition of the Laplace Transform 321 A2 The Inverse Laplace Transform 323 A3 Partial Fraction Expansion 323 Appendix B: The z-transform 329 Bl Definition of the z-transform 329 B2 The Inverse z-transform 330 B2.1 Inversion by Partial Fraction Expansion 330 Bibliography 335 Index 339

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