Lanczos(1, 2) has considered the shifted eigenvalue problem where C is a (p × q) matrix of rank r, CH is its hermitean conjugate and u, v are column vectors of orders p, q respectively. In this note we extend Lanczos' work to cover the more general eigenvalue problem which arises in certain problems in solid state physics (3). In (2), the I's are unit matrices of appropriate orders and the constants a, b are real; the partitioned matrix M is thus hermitean so that its eigenvalues λ are real.