Appendix A

Matrix Analysis

Finite-dimensional random vectors are the basic building blocks of many applications. Halmos (1958) [1472] is the standard reference. We just take the most elementary material from him.

Definition A.1 (Vector space)

*addition is communicative*,**x**+**y**=**y**+**x**,*addition is associative*,**x**+ (**y**+**z**) = (**x**+**y**) +**z**,*there exists in Ω a unique vector (called the origin) such that***x**+**0**=**x**,*for every***x**,*and**to every vector in Ω there corresponds a unique vector*−**x***such that***x**(−**x**) =**0**.

1. *multiplication by scalars is associative*, α(β**x**) = (αβ**x**), *and*

2. **1x** = **x**.

1. *Multiplication by scalars is distributive with respect to vector addition*, α(**x** + **y**) = α**x** + β**y**, *and*

2. *Multiplication by vectors is distributive with respect to scalar addition*, (α + β)**x** = α**x** + β**x**.

These axioms are not claimed to be logically independent.

Example A.1

1. Let be the set of all complex numbers; If we regard **x** + **y** and α**x** as ordinary complex numerical addition and multiplication, ...

Start Free Trial

No credit card required