Introduction to Matrices and Vectors

Preface
Chapter 1. Definition, Equality, and Addition of Matrices
1. Introduction
2. What Matrices Are
3. Equality of Matrices. Specification of Matrices. The Zero Matrix
4. Addition of Matrices
5. Addition of Matrices Continued
6. Addition of Matrices Concluded
7. Numerical Multiples of Matrices
Chapter 2. Multiplication of Matrices
1. A Problem Arising in Business Management
2. Formal Definition of Multiplication
3. A Surprising Property of Matrix Multiplication
4. Matrix Multiplication Continued
5. The Unit Matrix
6. The Laws of Matrix Multiplication
7. Powers of Matrices. Laws of Exponents
8. Multiplication of Matrices by Matrices and Multiplication of Matrices by Numbers
9. Polynomials in a Matrix
Chapter 3. Division of Matrices
1. Introduction
2. Using an Equation Satisfied by a Matrix
3. The Least Equation Satisfied by a Matrix
4. Using the Least Equation Satisfied by a Matrix to Solve the Problem of Reciprocals
5. Proof of the Uniqueness of the Least Equation Satisfied by a Matrix
6. Two Theorems about Reciprocals
Chapter 4. Vectors and Linear Equations
1. Definition of Vectors. Notation and Properties
2. Vectors and Directed Segments in the Plane
3. Vectors and Directed Segments in Three-dimensional Space
4. Geometric Applications of the Algebra of Vectors
5. Distances, Cosines, and Vectors
6. More about the Lengths of Vectors
7. Expressing Systems of Linear Equations in Terms of Vectors and Matrices. Solving by Using Reciprocal Matrices
8. Solving Systems of Linear Equations by the Method of Elimination
Chapter 5. Special Matrices of Particular Interest
1. The Complex Numbers as Real Matrices
2. The Quaternion Matrices
3. Matrices with Complex Entries
Chapter 6. More Algebra of Matrices and Vectors
1. The Transpose Matrix
2. The Trace of a Matrix
3. The Cross Product of Matrices
4. 3 x 3 Skew Matrices and Vectors of Size 3
5. Geometry of the Cap Product
Chapter 7. Eigenvalues and Eigenvectors
1. Matrix Reciprocals and Vectors
2. Eigenvalues
Chapter 8. Infinite Series of Matrices
1. The Geometric Series
2. The Size of the Entries in the Powers of A
3. The Exponential Series
Index