Minimization of a Scaled Tau Coefficient Using a Cyclic Coordinate Descent Algorithm

Abstract

Let P be a given nonnegative, irreducible matrix. A necessary and sufficient condition for the existence of a positive diagonal matrix D such that τ₁(D^-1PD) < τ₁(P) is obtained, where τ₁ is the tau coefficient defined with respect to the e¹ norm. We show that it is possible to obtain a smaller upper bound on the maximum modulus of the subdominant eigenvalues of P by minimizing τ₁ (D^-1PD) with respect to the entries in D using a cyclic coordinate descent algorithm.