Introduction

The multivariate Gaussian distribution is arguably the most important multivariate distribution in Digital Communications. It is the extension of the univariate Gaussian distribution from scalars to vectors. A random vector of this distribution is said to be a Gaussian vector, and its components are said to be jointly Gaussian. In this chapter we shall define this distribution, provide some useful characterizations, and study some of its key properties. To emphasize its connection to the univariate distribution, we shall derive it along the same lines we followed in deriving the univariate Gaussian distribution in Chapter 19.

There are a number of equivalent ways to define the multivariate Gaussian distribution, and authors typically pick one definition and then proceed over the course of numerous pages to derive alternate characterizations. We shall also proceed in this way, but to satisfy the impatient reader's curiosity we shall state the various equivalent definitions in this section. The proof of their equivalence will be spread over the whole chapter.

In the following definition we use the notation introduced in Section 17.2. In particular, all vectors are column vectors, and we denote the components of the vector a ∈ ℝn by a(1), …, a(n).