A Spectral Theoretic Proof of Perron–Frobenius
M. R.F. Symth · Mathematical Proceedings of the Royal Irish Academy · 2002
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