Abstract
This paper develops a fundamental result in linear algebra by providing the necessary and sufficient conditions for the simultaneous quasi-diagonalization of two symmetric matrices and two skew-symmetric matrices by a real orthogonal congruence. This result is used to study the uncoupling of general linear multi-degree-of-freedom (MDOF) structural and mechanical systems described by arbitrary damping and stiffness matrices through quasi-diagonalization, and real orthogonal coordinate transformations. The uncoupling leads to independent subsystems, each having at most two degrees of freedom with a specific structure. The results encompass the different physical categories of linear MDOF systems identified by engineers, mathematicians, and physicists, and provide the necessary and sufficient conditions for their maximal uncoupling. A total of sixteen conditions are shown to exist. However, the number of such conditions for physical systems that are commonly met in nature as well as in aerospace, civil, and mechanical engineering are shown to be considerably less, dwindling at times to two or three, thereby making the results applicable to numerous high-order real-life linear MDOF dynamical systems. Several new analytical results are obtained and corroborated through numerical examples.
