Appendix A

Matrix Analysis

# A.1 Vector Spaces and Hilbert Space

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.

*A vector space is a set Ω of elements called vectors satisfying the following axioms. (A) To every pair*

**x**

*and*

**y**,

*of vectors in Ω, there corresponds a vector*

**x**+

**y**,

*called the sum of*

**x**

*and*

**y**,

*in such a way that*

*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**.

*(B) To every pair, α and*

**x**,

*where α is a scalar and*

**x**

*is a vector in Ω, there corresponds a vector*α

**x**

*in Ω, called the product of α and*

**x**,

*in such a way that*

*multiplication by scalars is associative*, α(β

**x**) = (αβ

**x**),

*and*

**1x**=

**x**.

*(C)*

*Multiplication by scalars is distributive with respect to vector addition*, α(

**x**+

**y**) = α

**x**+ β

**y**,

*and*

*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.

**x**+

**y**and α

**x**as ordinary complex numerical addition and multiplication, ...

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