Preliminaries

Abstract

Denote by R the set of real numbers and Rn their n-fold Cartesian product R × … × R, the set of all ordered n-tuples (x¹, …, xⁿ). Define a function

$$ d:R^n \times R^n ,{\text{ where }}d(x,y) = {\text{ }}\parallel x - y\parallel $$

for every pair (x, y) of the points x, y ∈ Rⁿ. This function d is known as the Euclidean metric in Rⁿ. Then, we call Rⁿ with the metric d the n-dimensional Euclidean space. Consider V a real n-dimensional vector space with a symmetric bilinear mapping g: V × V → R. We say that g is positive (negative) definite on V if g(v, v) ≥ 0 (g(v, v) ≤ 0) for any non-zero v ∈ V. On the other hand, if g(v, v)=0 (g(v, v) ≤ 0) for any v ∈ V and there exists a non-zero u ∈ V with g(u, u)=0, we say that g is positive (negative) semi-definite on V.