## Book description

This book is intended as an undergraduate text introducing matrix methods as they relate to engineering problems. It begins with the fundamentals of mathematics of matrices and determinants. Matrix inversion is discussed, with an introduction of the well known reduction methods. Equation sets are viewed as vector transformations, and the conditions of their solvability are explored. Orthogonal matrices are introduced with examples showing application to many problems requiring three dimensional thinking. The angular velocity matrix is shown to emerge from the differentiation of the 3-D orthogonal matrix, leading to the discussion of particle and rigid body dynamics.

The book continues with the eigenvalue problem and its application to multi-variable vibrations. Because the eigenvalue problem requires some operations with polynomials, a separate discussion of these is given in an appendix. The example of the vibrating string is given with a comparison of the matrix analysis to the continuous solution.

Table of Contents: Matrix Fundamentals / Determinants / Matrix Inversion / Linear Simultaneous Equation Sets / Orthogonal Transforms / Matrix Eigenvalue Analysis / Matrix Analysis of Vibrating Systems

1. Preface
2. Matrix Fundamentals
1. Definition of A Matrix
2. Elemetary Matrix Algebra
3. Basic Types of Matrices
4. Transformation Matrices
5. Matrix Partitioning
6. Interesting Vector Products
7. Examples
8. Exercises
3. Determinants
1. Introduction
2. General Definition of a Determinant
3. Permutations and Inversions of Indices
4. Properties of Determinants
5. The Rank of a Determinant
6. Minors and Cofactors
7. Geometry: Lines, Areas, and Volumes
8. The Adjoint and Inverse Matrices
9. Determinant Evaluation
10. Examples
11. Exercises
4. Matrix Inversion
1. Introduction
2. Elementary Operations in Matrix Form
3. Gauss-Jordan Reduction
4. The Gauss Reduction Method
5. LU Decomposition
6. Matrix Inversion By Partitioning
8. Examples
9. Exercises
5. Linear Simultaneous Equation Sets
1. Introduction
2. Vectors and Vector Sets
3. Simultaneous Equation Sets
4. Linear Regression
5. Lagrange Interpolation Polynomials
6. Exercises
6. Orthogonal Transforms
1. Introduction
2. Orthogonal Matrices and Transforms
3. Example Coordinate Transforms
4. Congruent and Similarity Matrix Transforms
5. Differentiation of Matrices, Angular Velocity
6. Dynamics of a Particle
7. Rigid Body Dynamics
8. Examples
9. Exercises
7. Matrix Eigenvalue Analysis
1. Introduction
2. The Eigenvalue Problem
3. Geometry of the Eigenvalue Problem
4. The Eigenvectors and Orthogonality
5. The Cayley-Hamilton Theorem
6. Mechanics of the Eigenvalue Problem
7. Example Eigenvalue Analysis
8. The Eigenvalue Analysis of Similar Matrices; Danilevsky's Method
9. Exercises
8. Matrix Analysis of Vibrating Systems
1. Introduction
2. Setting up Equations, Lagrange's Equations
3. Vibration of Conservative Systems
4. Nonconservative Systems. Viscous Damping
6. Runge-Kutta Integration of Differential Equations
7. Exercises
9. Partial Differentiation of Bilinear and Quadratic Forms
10. Polynomials
1. Polynomial Basics
2. Polynomial Arithmetic
3. Evaluating Polynomial Roots
11. The Vibrating String
12. Solar Energy Geometry