In connection with previously published work, this paper presents further results about the bounding properties of eigenvalues provided by a linear eigenmatrix formulation A - lambda B. The linear eigenmatrix is formed by expressing the elements of a non-linear dynamic stiffness matrix, K(lambda), as linear functions of the eigenparameter lambda. This is achieved by choosing two fixed values of the eigenparameter and calculating K(lambda) at these two values. The eigenvalues of A - lambda B provide error bounds on the exact eigenvalues of the non-linear eigenmatrix if the two fixed values chosen are below the lowest pole of K(lambda). Choosing two identical fixed values, the error bounds on the exact eigenvalues provided by traditional linearization techniques are found as special cases. (C) 1998 John Wiley & Sons, Ltd.