Paths of matrices with the strong Perron-Frobenius property converging to a given matrix with the Perron-Frobenius property

Abstract

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius property such that ∥A−A(ε)∥ ≤ε. In this note, the form that the parameterized matrices A(ε) and their spectral characteristics can take are studied. It is shown to be possible to have A(ε) cubic, its spectral radius quadratic and the corresponding positive eigenvector linear (all as functions of ε); further, if the spectral radius of A is simple, positive and strictly dominant, then A(ε) can be taken to be quadratic and its spectral radius linear (in ε). Two other cases are discussed: when A is normal it is shown that the sequence of approximating matrices A(ε) can be written as a quadratic polynomial in trigonometric functions, and when A has semipositive left and right Perron-Frobenius eigenvectors and ρ(A) is simple, the sequence A(ε) can be represented as a polynomial in trigonometric functions of degree at most six.