Complex Networks: A Networking and Signal Processing Perspective

Appendix AVectors and Matrices

A.1 Vectors and Norms

A vector is an array of real or complex numbers. We denote vectors by lowercase bold letters and assume a vector to be a column vector. For example, a vector of length n is

𝐱=[x1x2⋮xn].

The number of elements in a vector is also known as the dimension of the vector. If the elements x₁, x₂,... , x_n are real, then we say that x is a real n-dimensional vector, and we can denote it as 𝐱∈ℝn. On the other hand, if the elements are complex, then we say that x is a complex n-dimensional vector, and it is denoted as 𝐱∈ℂn.

Various norms are defined to measure the magnitude of a vector. One such measure is ℓ_p norm, which is defined as

(A.1.1) ||X||p=[Σi=1n|xi|p]1/p,

where n is the dimension of ...