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

Chapter 1Basic properties of vectors and matrices

1 INTRODUCTION

In this chapter, we summarize some of the well‐known definitions and theorems of matrix algebra. Most of the theorems will be proved.

2 SETS

A set is a collection of objects, called the elements (or members) of the set. We write x ∈ S to mean ‘x is an element of S’ or ‘x belongs to S’. If x does not belong to S, we write x ∉ S. The set that contains no elements is called the empty set, denoted by ∅.

Sometimes a set can be defined by displaying the elements in braces. For example, A = {0, 1} or

Notice that A is a finite set (contains a finite number of elements), whereas ℕ is an infinite set. If P is a property that any element of S has or does not have, then

denotes the set of all the elements of S that have property P.

A set A is called a subset of B, written A ⊂ B, whenever every element of A also belongs to B. The notation A ⊂ B does not rule out the possibility that A = B. If A ⊂ B and A ≠ B, then we say that A is a proper subset of B.

If A and B are two subsets of S, we define

the union of A and B, as the set of elements of S that belong to A or to B or to both, and

the intersection of A and B, as the set of elements ...