Summary: Kron’s primitive composite system matrix is shown to be reducible to a symmetric indefinite matrix. This matrix, although never formed explicitly, can then be transformed to a tridiagonal matrix of reduced order by the Lanczos algorithm. Eigenvalue solutions of this partially tridiagonalized matrix give very good approximations to the eigenvalues of the composite system. The method has multiplication counts which are over 90 per cent lower than when the (condensed) Kron matrix is solved by the Newton-Raphson iteration applied to the Kron scalar equation. Numerical examples illustrating the Kron-Lanczos method when solving for the natural frequencies of frames with large numbers of degrees of freedom are presented.