Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 11Inequalities

1 INTRODUCTION

Inequalities occur in many disciplines. In economics, they occur primarily because economics is concerned with optimizing behavior. In other words, we often wish to find an x^* such that ϕ(x^*) ≥ ϕ(x) for all x in some set. The equivalence of the inequality ϕ(x) ≥ 0 for all x ∈ S and the statement

suggests that inequalities can often be proved by reformulating the inequality as a minimization problem and then use differential calculus. We shall see in this chapter that this method does not always lead to success, but if it does we shall use it.

The chapter falls naturally into several parts. In Sections 11.1–11.4, we discuss (matrix analogs of) the Cauchy‐Schwarz inequality and the arithmetic‐geometric means inequality. Sections 11.5–11.14 are devoted to inequalities concerning eigenvalues and contain inter alia Fischer's min‐max theorem and Poincaré's separation theorem. In Section 11.15, we prove Hadamard's inequality. In Sections 11.16–11.23, we use Karamata's inequality to prove a representation theorem for (tr A^p)^1/p with p > 1 and A positive semidefinite, which in turn is used to establish matrix analogs of the inequalities of Hölder and Minkowski. Sections 11.24 and 11.25 contain Minkowski's determinant theorem. In Sections 11.26–11.28, several inequalities concerning the weighted means of order p are discussed. Finally, in Sections 11.29–11.32, ...