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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)  is the standard reference. We just take the most elementary material from him.

Definition A.1 (Vector space)
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 vectorx 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 αx in Ω, called the product of α and x, in such a way that
1. multiplication by scalars is associative, α(βx) = (αβx), and
2. 1x = x.
(C)
1. Multiplication by scalars is distributive with respect to vector addition, α(x + y) = αx + βy, and
2. Multiplication by vectors is distributive with respect to scalar addition, (α + β)x = αx + βx.

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 αx as ordinary complex numerical addition and multiplication, ...

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