### 1.7. Random Vector and Matrix Generation

For various simulation or power studies, it is often necessary to generate a set of random vectors or random matrices. It is therefore of interest to generate these quantities for the probability distributions which arise naturally in the multivariate normal theory. The following sections consider the most common multivariate probability distributions.

#### 1.7.1. Random Vector Generation from N_{p}(μ, Σ)

To generate a random vector from N_{p}(μ, Σ) use the following steps:

Find a matrix G such that Σ = G′G. This is obtained using the Cholesky decomposition of the symmetric matrix Σ. The functions ROOT of Half in PROC IML can perform this decomposition.

Generate p independent standard univariate normal random variables ...