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

15A48

Perron-Frobenius property, generalization of nonnegative matrices, eventually nonnegative matrices, eventually positive matrices, perturbation