MATRIX THEORY & APPLICATIONS FOR SCIENTISTS & ENGINEERS

Description

This book steers a middle course between the two extremes. With the rapid use of even more sophisticated techniques in science, economics and engineering, and especially in control theory, it has become essential to have a real understanding of the many methods being used. To achieve simplicity combined with as deep an understanding as possible, I have tried to use very simple notation, even at the possible expense of rigour. I know that a student who, for example, is not used to Gothic lettering, will actually find a passage of mathematics using such symbols (to denote sets) much more difficult to absorb than when Roman lettering is used.

Table of Contents
Preface
Chapter 1 – Matrices
1.1 Matrices and Matrix Operations
1.2 Some Properties of Matrix Operations
1.3 Partitioned Matrices
1.4 Some Special Matrices
1.5 The State – Space Concept
Chapter 2 – Vector Spaces
2.1 Vectors
2.2 Linear Dependence and Bases
2.3 Coordinates and the Transition Matrix
Chapter 3 – Linear Transformations
3.1 Homomorphisms
3.2 Isomorphism and Vector Spaces
3.3 Linear Transformations and Matrices
3.4 Orthogonal Transformations
3.5 General Change of Bases for a Linear Transformation
Chapter 4 – The Rank and the Determinant of a Matrix
4.1 The Kernel and the Image Space of a Linear Transformation
4.2 The Rank of a Matrix
4.3 The Determinant of a Matrix
4.4 Operations with Determinants
4.5 Cramer’s Rule
Chapter 5 – Linear Equations
5.1 Systems of Homogeneous Equations
5.2 Systems of Non-Homogeneous Equations
Chapter 6 – Eigenvectors and Eigenvalues
6.1 The Characteristic Equation
6.2 The Eigenvalues of the transposed matrix
6.3 When all the Eigenvalues of A are distinct
6.4 A reduction to a Diagonal Form
6.5 Multiple Eigenvalues
6.6 The Cayley-Hamilton Theorem
Chapter 7 – Canonical Forms and Matrix Functions
7.1 Polynomials
7.2 Eigenvalues of Rational Functions of a Matrix
7.3 The Minimum Polynomial of a Matrix
7.4 Direct Sums and Invariant Subspaces
7.5 A Decomposition of a Vector Space
7.6 Cyclic Bases and the Rational Canonical Form
7.7 The Jordan Canonical Forms
7.8 Matrix Functions
Chapter 8 – Inverting a Matrix
8.1 Elementary Operations and Elementrry Matrices
8.2 The Inverse of a Vandermonde Matrix
8.3 Faddeeva’s Method
8.4 Inverting a Matrix with Complex Elements
Solutions to Problems
References and Bibliography
Index