book

Linear Algebra and Matrix Computations with MATLAB®

Name: Linear Algebra and Matrix Computations with MATLAB®
ISBN: 9783110663716

by Dingyü Xue, Tsinghua University Tsinghua University Press

March 2020

Intermediate to advanced

301 pages

8h 23m

English

De Gruyter

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Preface
1 Introduction to linear algebra and matrix computation
1.1 Matrices and linear equations1.1.1 Matrix representation of tables1.1.2 Modeling and solving linear equations1.2 Introduction of linear algebra1.2.1 Mathematical development of linear algebra1.2.2 Numerical linear algebra1.3 Outline of the book1.4 Exercises of the chapter
2 Representation and fundamental computations of matrices
2.1 Input of an ordinary matrix2.2 Input of special matrices2.2.1 Matrices of zeros, ones, and identity matrices2.2.2 Matrices with random elements2.2.3 Hankel matrix2.2.4 Diagonal matrices2.2.5 Hilbert matrix and its inverse2.2.6 Companion matrices2.2.7 Vandermonde matrices2.2.8 Some commonly used test matrices2.3 Symbolic matrix input2.3.1 Input of special symbolic matrices2.3.2 Generating arbitrary constant matrices2.3.3 Arbitrary matrix function input2.4 Input of sparse matrices2.5 Fundamental matrix computations2.5.1 Handling complex matrices2.5.2 Transposition and rotation of matrices2.5.3 Algebraic computation of matrices2.5.4 Kronecker products and sums2.6 Calculus computations with matrices2.6.1 Matrix derivatives2.6.2 Integrals of matrix functions2.6.3 Jacobian matrices of vector functions2.6.4 Hessian matrices2.7 Exercises
3 Fundamental analysis of matrices
3.1 Determinants3.1.1 Determinant definition and properties3.1.2 Determinants of small-scale matrices3.1.3 MATLAB solution of determinant problems3.1.4 Determinants of special matrices of any sizes3.1.5 Cramer’s rule in linear equations3.1.6 Positive and totally positive matrices3.2 Simple analysis of matrices3.2.1 The traces3.2.2 Linear independence and matrix rank3.2.3 Norms3.2.4 Vector space3.3 Inverse and generalized inverse matrices3.3.1 Inverse matrices3.3.2 Derivatives of inverse matrix3.3.3 MATLAB based inverse matrix evaluation3.3.4 Reduced row echelon form3.3.5 Generalized inverse3.4 Characteristic polynomials and eigenvalues3.4.1 Characteristic polynomials3.4.2 Finding the roots of polynomial equations3.4.3 Eigenvalues and eigenvectors3.4.4 Generalized eigenvalues and eigenvectors3.4.5 Gershgorin discs and diagonally dominant matrices3.5 Matrix polynomials3.5.1 Solutions of matrix polynomial problems3.5.2 Minimal polynomials of matrices3.5.3 Conversions between symbolic and numerical polynomials3.6 Problems
4 Fundamental transformation and decomposition of matrices
4.1 Similarity transform and orthogonal matrices4.1.1 Similarity transform4.1.2 Orthogonal matrices and orthonormal bases4.2 Elementary row transforms4.2.1 Three types of elementary row transforms4.2.2 Inverse through elementary row transforms4.2.3 Computing inverses with pivot methods4.3 Triangular decomposition of matrices4.3.1 Gaussian elimination of linear equations4.3.2 Triangular factorization and implementation4.3.3 MATLAB triangular factorization solver4.4 Cholesky factorization4.4.1 Cholesky factorization of symmetric matrices4.4.2 Quadratic forms of symmetric matrices4.4.3 Positive definite and regular matrices4.4.4 Cholesky factorization of nonpositive definite matrices4.5 Companion and Jordan transforms4.5.1 Companion form transformation4.5.2 Matrix diagonalization4.5.3 Jordan transform4.5.4 Real Jordan transforms for matrices with repeated complex eigenvalues4.5.5 Simultaneous diagonalization4.6 Singular value decomposition4.6.1 Singular values and condition numbers4.6.2 Singular value decomposition of rectangular matrices4.6.3 SVD based simultaneous diagonalization4.7 Givens and Householder transforms4.7.1 Two-dimensional coordinate rotation transform4.7.2 Givens transform of ordinary matrices4.7.3 Householder transform4.8 Problems
5 Solutions of matrix equations
5.1 Linear equations5.1.1 Unique solutions5.1.2 Linear equations with infinitely many solutions5.1.3 Inconsistent systems of equations5.1.4 Graphical interpretation of linear solutions5.2 Other forms of simple linear equations5.2.1 XA=B equations5.2.2 AXB=C equations5.2.3 Kronecker product-based solutions5.2.4 Multiterm AXB=C equations5.3 Lyapunov equations5.3.1 Continuous Lyapunov equations5.3.2 2×2 Lyapunov equations5.3.3 Analytical solution of Lyapunov equations5.3.4 Stein equations5.3.5 Discrete Lyapunov equation5.4 Sylvester equations5.4.1 Numerical solutions of Sylvester equations5.4.2 Analytical solution of Sylvester equation5.4.3 Sylvester equations with constant parameters5.4.4 Multiterm Sylvester equations5.5 Nonlinear matrix equations5.5.1 Riccati equations5.5.2 Solutions of general nonlinear equations5.5.3 Variations of Riccati equations5.5.4 Numerical solutions of nonlinear matrix equations5.6 Diophantine equations5.6.1 Coprimeness of polynomials5.6.2 Diophantine polynomial equations5.6.3 Solutions of pseudo-polynomial equations5.7 Problems
6 Matrix functions
6.1 Nonlinear computation of matrix elements6.1.1 Integer rounding and rationalization6.1.2 Transcendental function computations6.1.3 Vector sorting, minimization, and maximization6.1.4 Means, variances, and standard deviations6.2 Matrix exponentials6.2.1 Definitions and properties of matrix functions6.2.2 Computing matrix exponentials6.2.3 Taylor series-based truncation algorithm6.2.4 A Cayley–Hamilton theorem-based algorithm6.2.5 Direct MATLAB solver6.2.6 Jordan transform-based approach6.3 Matrix logarithmic and square root functions6.3.1 Matrix logarithmic function6.3.2 Square root of a matrix6.4 Matrix trigonometric functions6.4.1 Computation of matrix trigonometric functions6.4.2 Power series-based trigonometric function evaluation6.4.3 Analytical evaluations of matrix trigonometric functions6.5 Arbitrary matrix functions6.5.1 Nilpotent matrices6.5.2 Jordan transform-based matrix function evaluations6.5.3 User-defined matrix functions6.6 Matrix powers6.6.1 Matrix powers via Jordan transform6.6.2 A matrix power solver6.6.3 Matrix powers via the z transform6.6.4 Computing kA6.7 Problems
7 Linear algebra applications
7.1 Linear equations applications7.1.1 Electric network analysis7.1.2 Analysis of truss structures7.1.3 Balancing chemical reaction equations7.2 Applications in linear control systems7.2.1 Control system model conversions7.2.2 Qualitative analysis of linear systems7.2.3 Transmission zeros in multivariate systems7.2.4 Direct solutions of linear differential equations7.3 Introduction to digital image processing7.3.1 Image input and display7.3.2 Singular value decomposition applications7.3.3 Resizing and rotating images7.3.4 Image enhancement7.4 Graph theory and applications7.4.1 Oriented graph descriptions7.4.2 Dijkstra algorithm and applications7.4.3 Simplification of control system block diagrams7.5 Solving difference equations7.5.1 Analytical solutions7.5.2 Numerical solutions of linear time-varying difference equations7.5.3 Linear time-invariant difference equations7.5.4 Numerical solutions of nonlinear difference equations7.5.5 Markov chain simulation7.6 Data fitting and analysis7.6.1 Linear regression7.6.2 Polynomial fitting7.6.3 Chebyshev polynomials7.6.4 Bézier curves7.6.5 Principal component analysis7.7 Exercises of the chapter
Bibliography
Subject Index

Sachregister

Content preview from Linear Algebra and Matrix Computations with MATLAB®

6 Matrix functions

Matrix functions are very important in matrix analysis. They play important roles in many scientific and engineering applications.

The term “matrix function” may have different meanings in mathematics^[12]. In this chapter, the following definition is mainly used.

Definition 6.1.

Given a scalar function $f(x)$ and a complex square matrix $A$ , if the independent variable x is substituted by matrix $A$ , $f(A)$ is a matrix of the same size, then $f(A)$ is referred to as a matrix function.

It can be seen that matrix function is an extension of scalar function. It is itself a function, and can be used to map one matrix into another through matrix function. An example below is given to demonstrate the extension from scalar functions. ...

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Linear Algebra and Matrix Computations with MATLAB®

by Dingyü Xue, Tsinghua University Tsinghua University Press

6 Matrix functions

Definition 6.1.

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