2002/01/01 by M. R.F. Symth · 1 citation
Computer Science · Engineering · Mathematics · #Matrix Theory and Algorithms #graph theory and CDMA systems #Graph theory and applications #Mathematics #Eigenvalues and eigenvectors #Spectral radius #Spectrum (functional analysis) #Combinatorics #Projection (relational algebra) #Matrix (chemical analysis) #Operator (biology) #Generalized eigenvector #Diagonal #Vector space #Pure mathematics #Discrete mathematics #Symmetric matrix #Geometry #Algorithm
paper · doi:10.3318/pria.2002.102.1.29
openalex publication_date 2002/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
An elementary proof of the famous Perron–Frobenius Theorem for non-negative matrices is given via classical spectral theory. The key to the proof lies in the inherited characteristics of certain spectral projections. 1. Preliminaries In this paper T will be a linear operator on a complex finite-dimensional vector space. We shall formally denote its matrix relative to a fixed basis by [T ] and the actual element in row i, column j by [T ]ij. Where there is no danger of confusion the square brackets are omitted. The main diagonal vector of [T ] is denoted by diag(T ). The spectrum or set of eigenvalues of T is denoted by σ(T ) and the largest modulus amongst its eigenvalues, the spectral radius of T , is denoted by r(T ). We use P (λ; T ) to denote the spectral projection associated with the eigenvalue λ relative to the operator T . The peripheral spectrum π(T ) is the set of eigenvalues of T with modulus r(T ), and the number of elements in π(T ) is called the index of imprimitivity