3
Orthogonality
In this chapter, we will analyze distance functions, inner products, projection
and orthogonality, the process of finding an orthonormal basis, QR and sin-
gular value decompositions and conclude with a final discussion about how to
solve the general form of Ax = b.3.1 Inner Products
Following a rapid review of norms, an operation between any two vectors
in the same space, inner product, is discussed together with the associated
geometric implications.3.1.1 Norms
Norms (distance functions, metrics) are vital in characterizing the type of
network optimization problems like the Travelling Salesman Problem (TSP)
with the rectilinear distance.Definition 3.1.1 A norm on a vector space V is a function that assigns to
each vector, v € V, a nonnegative real number \v\ satisfying
i. \v\ >0,Vvy£9 and \6\ = 0,
ii. \av\ - \a\ \v\, Ma € K; v £ V.
Hi. \u + v\ < \u\ + \v\, Vu, v € V (triangle inequality).
Definition 3.1.2 Vrc G Cn, the most commonly used norms, H-l^ , ||.|| 2 , H-H^,
are called the li, li and l^ norms, respectively. They are defined as below:
- \x\x = |xi| + --- + |arn|,
- ||x|| 2 = (|x 1 |^2 + --- + |o;n|^2 )i;
(^3) - Halloo z=max{la;l|)--->l;En|}-