A direct approach to second-order matrix non-classical vibrating equations Original Research Article

A direct framework is developed for second-order matrix equations without transforming them into a first-order companion equation. It is done in terms of its matrix impulse response that is directly related to the transfer matrix. We formulate an extension of the Cayley–Hamilton identity, derive the controllability and observability matrices, and discuss Krylovʹs method in terms of such matrix response. This formulation will allow to further discuss Arnoldi and Lanczos methods as well as time-integration by FFT.