Linear algebra preliminaries

Abstract

With the notations R ⁿ and R ^m×n we denote respectively the space of n-vectors and m × n matrices with coefficients in a ring R — or a semi-ring, as in the case of

$${\mathbb{R}_ + }: = \left\{ {x \in \mathbb{R}:x \geqslant 0} \right\}.$$

The notation I_n, with n often omitted when it is clear from the context, denotes the identity matrix; the zero matrix of any dimension is denoted simply by 0. With e we denote the vector of suitable dimension all of whose entries are 1. The expression ρ (A) stands for the spectral radius of the matrix A.