Matrix Methods

CHAPTER 1 Matrices

1.1 Basic concepts
Problems 1.1

1.2 Operations
Problems 1.2

1.3 Matrix multiplication
Problems 1.3

1.4 Special matrices
Problems 1.4

1.5 Submatrices and partitioning
Problems 1.5

1.6 Vectors
Problems 1.6

1.7 The geometry of vectors
Problems 1.7
CHAPTER 2 Simultaneous linear equations

2.1 Linear systems
Problems 2.1

2.2 Solutions by substitution
Problems 2.2

2.3 Gaussian elimination
Problems 2.3

2.4 Pivoting strategies
Problems 2.4

2.5 Linear independence
Problems 2.5

2.6 Rank
Problems 2.6

2.7 Theory of solutions
Problems 2.7

2.8 Final comments on Chapter 2
CHAPTER 3 The inverse

3.1 Introduction
Problems 3.1

3.2 Calculating inverses
Problems 3.2

3.3 Simultaneous equations
Problems 3.3

3.4 Properties of the inverse
Problems 3.4

3.5 LU decomposition
Problems 3.5

3.6 Final comments on Chapter 3
CHAPTER 4 An introduction to optimization

4.1 Graphing inequalities
Problems 4.1

4.2 Modeling with inequalities
Problems 4.2

4.3 Solving problems using linear programming
Problems 4.3

4.4 An introduction to the simplex method
Problems 4.4

4.5 Final comments on Chapter 4
CHAPTER 5 Determinants

5.1 Introduction
Problems 5.1

5.2 Expansion by cofactors
Problems 5.2

5.3 Properties of determinants
Problems 5.3

5.4 Pivotal condensation
Problems 5.4

5.5 Inversion
Problems 5.5

5.6 Cramer’s rule
Problems 5.6

5.7 Final comments on Chapter 5
CHAPTER 6 Eigenvalues and eigenvectors

6.1 Definitions
Problems 6.1

6.2 Eigenvalues
Problems 6.2

6.3 Eigenvectors
Problems 6.3

6.4 Properties of eigenvalues and eigenvectors
Problems 6.4

6.5 Linearly independent eigenvectors
Problems 6.5

6.6 Power methods
Problems 6.6
CHAPTER 7 Matrix calculus

7.1 Well-defined functions
Problems 7.1

7.2 Cayley-Hamilton theorem
Problems 7.2

7.3 Polynomials of matricesddistinct eigenvalues
Problems 7.3

7.4 Polynomials of matricesdgeneral case
Problems 7.4

7.5 Functions of a matrix
Problems 7.5

7.6 The function eAt
Problems 7.6

7.7 Complex eigenvalues
Problems 7.7

7.8 Properties of eA
Problems 7.8

7.9 Derivatives of a matrix
Problems 7.9

7.10 Final comments on Chapter 7
CHAPTER 8 Linear differential equations

8.1 Fundamental form
Problems 8.1

8.2 Reduction of an nth order equation
Problems 8.2

8.3 Reduction of a system
Problems 8.3

8.4 Solutions of systems with constant coefficients
Problems 8.4

8.5 Solutions of systemsdgeneral case
Problem 8.5

8.6 Final comments on Chapter 8
CHAPTER 9 Probability and Markov chains

9.1 Probability: an informal approach
Problems 9.1

9.2 Some laws of probability
Problems 9.2

9.3 Bernoulli trials and combinatorics
Problems 9.3

9.4 Modeling with Markov chains: an introduction
Problems 9.4

9.5 Final comments on Chapter 9
CHAPTER 10 Real inner products and least square

10.1 Introduction
Problems 10.1

10.2 Orthonormal vectors
Problems 10.2

10.3 Projections and QR decompositions
Problems 10.3

10.4 The QR algorithm
Problems 10.4

10.5 Least squares
Problems 10.5
CHAPTER 11 Sabermetrics e An introduction

11.1 Introductory comments

11.2 Some basic measures

11.3 Sabermetrics in the classroom

11.4 Run expectancy matrices

11.5 How to “do” sabermetrics

11.6 Informal reference list

11.7 Testing
CHAPTER 12 Sabermetrics e A module

12.1 Base stealing runs (BSRs)

12.2 Batting linear weights runs (BLWTS)

12.3 Equivalence coefficient (EC)

12.4 Isolated power (ISO)

12.5 On base average (OBA)

12.6 On base plus slugging (OPS)

12.7 Power factor (PF)

12.8 Power-speed number (PSN)

12.9 Runs created (RC)

12.10 Slugging times on base average (SLOB)

12.11 Total power quotient (TPQ)

12.12 Modified weighted pitcher’s rating (MWPR)

12.13 Pitching linear weights runs (PLWTS)

12.14 Walks plus hits per innings pitched (WHIP)
Appendix: A word on technology
Answers and hints to selected problems