We give a formal definition of preconditioned eigensolvers as polynomial methods for generalized symmetric eigenvalue problems. We consider our new preconditioned conjugate gradient method, and argue that this is a genuine conjugate gradient method by comparing it with the preconditioned conjugate gradient method for linear systems of equations and with the total
minimization method.

The talk is partially based on the papers: "Preconditioned eigensolvers - an oxymoron?" , ETNA, 7 (1998), pp. 104-123, and Preconditioned eigensolvers: practical algorithms, Published as a technical report UCD-CCM 143, 1999, at the Center for Computational Mathematics, University of Colorado Denver. An edited version was accepted as a section to the project ``Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide,'' Editors: Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk Van der Vorst, SIAM, 1999.