A COOK-BOOK OF MATHEMATICS

Viatcheslav VINOGRADOV
June 1999

Contents
1 Linear Algebra
1
1.1 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1.1 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1.2 Laws of Matrix Operations . . . . . . . . . . . . . . . . . . . . . . .2
1.1.3 Inverses and Transposes . . . . . . . . . . . . . . . . . . . . . . . .3
1.1.4 Determinants and a Test for Non-Singularity . . . . . . . . . . . . .4
1.1.5 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
1.2 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . .8
1.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Appendix: Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.2 Vector Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.3 Independence and Bases . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.4 Linear Transformations and Changes of Bases . . . . . . . . . . . . 18
2 Calculus 21
2.1 The Concept of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Di?erentiation - the Case of One Variable . . . . . . . . . . . . . . . . . . 22
2.3 Rules of Di?erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Maxima and Minima of a Function of One Variable . . . . . . . . . . . . . 26
2.5 Integration (The Case of One Variable) . . . . . . . . . . . . . . . . . . . . 29
2.6 Functions of More than One Variable . . . . . . . . . . . . . . . . . . . . . 32
2.7 Unconstrained Optimization in the Case of More than One Variable . . . . 34
2.8 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Concavity, Convexity, Quasiconcavity and Quasiconvexity . . . . . . . . . . 38
2.10 Appendix: Matrix Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Constrained Optimization 43
3.1 Optimization with Equality Constraints . . . . . . . . . . . . . . . . . . . . 43
3.2 The Case of Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Non-Linear Programming . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Kuhn-Tucker Conditions . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Appendix: Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 The Setup of the Problem . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Dynamics 63
4.1 Di?erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Di?erential Equations of the First Order . . . . . . . . . . . . . . . 63
4.1.2 Linear Di?erential Equations of a Higher Order with Constant Co-
e?cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.3 Systems of the First Order Linear Di?erential Equations . . . . . . 68
4.1.4 Simultaneous Di?erential Equations. Types of Equilibria . . . . . . 77
4.2 Di?erence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1 First-order Linear Di?erence Equations . . . . . . . . . . . . . . . . 80
4.2.2 Second-Order Linear Di?erence Equations . . . . . . . . . . . . . . 82
4.2.3 The General Case of Order n . . . . . . . . . . . . . . . . . . . . . 83
4.3 Introduction to Dynamic Optimization . . . . . . . . . . . . . . . . . . . . 85
4.3.1 The First-Order Conditions . . . . . . . . . . . . . . . . . . . . . . 85
4.3.2 Present-Value and Current-Value Hamiltonians . . . . . . . . . . . 87
4.3.3 Dynamic Problems with Inequality Constraints . . . . . . . . . . . 87
5 Exercises 93
5.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 More Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Sample Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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