Lower bounds for the largest eigenvalue of a symmetric matrix under perturbations of rank one

Let A=(a(ij)) denote a real symmetric matrix of order n and define u as the vector of all ones in R(n). A classical lower bound for the largest eigenvalue of A is given by y(t)Ay/y(T)y with y=u. Some new developments were recently suggested by Walker and Van Mieghem [S. G. Walker and P. Van Mieghem, On lower bounds for the largest eigenvalue of a symmetric matrix, Linear Algebra Appl. 429 (2008), pp. 519-526] by applying this classical bound to suitable transforms of A. In this short note, it is shown that the approach of Walker and Van Mieghem can be used with choices other than y=u in order to improve classical inequalities when considering matrix perturbations of rank one.