Approximations of the Spectral Radius Corresponding Eigenvector, and Second Largest Modulus of an Eigenvalue for Square, Nonnegative, Irreducible Matrices

Abstract

Approximations are obtained for the normalized eigenvector of a square, nonnegative, irreducible matrix corresponding to its spectral radius from solutions to linear inequality systems whose feasibility have long been used to characterize lower and/or upper bounds on the spectral radius. These linear inequality systems depend on a parameter that can be viewed as an estimator for the spectral radius. In particular, we derive bounds on the tightness of the resulting approximation to the corresponding eigenvector as a product of a constant (equaling the optimal objective value of a nonlinear, convex optimization problem) and the difference between the spectral radius and its estimator. Bounds are also developed on the second largest modulus of an eigenvalue of a square, nonnegative, irreducible matrix in terms of approximations to its spectral radius and the corresponding normalized eigenvectors. The latter results depend on the methods of Rothblum and Tan [Linear Algebra Appl., 66 (1985), pp. 45–86], who derived bounds on the second largest modulus of an eigenvalue, which depends on the explicit knowledge of the spectral radius and corresponding eigenvector of the underlying matrix.