Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

10.5 Annex 5 – Detailed Mathematical Demonstrations

10.5.1 Basic results About Vector Random Variables and Matrices

Recall the following basic vector probabilistic results.

Considering X, as a p-dimensional random vector, and H, a q∗p matrix generating Y (of dimension q), then:

Considering any r∗s matrix denoted as D, then the symmetric matrix DD′ is also semi-positive definite. Indeed, for any vector y of dimension s, then:

The following property is useful in the context of model identifiability (Chapter 6):

H is a full-rank (i.e. injective) q∗p matrix  H′H is an invertible p∗p matrix.

Proof

Suppose H is injective, then:

which proves that H′H is also injective. Being a p∗p matrix, it is invertible.

Conversely, if H′H is invertible then:

which proves that H has a void kernel, that is, it is injective.

This property still holds if H′RH is invertible, R being a definite-positive q∗q matrix. Indeed, R may be decomposed under Cholesky as a product of a q∗q matrix and its transpose: