The concepts of non-selfadjointness are re-examined by differentiating between systems whose physical symmetries, in the sense of the Betti-Maxwell reciprocity theorem, are still preserved but are mathematically asymmetric, and systems that do not possess any physical symmetries. Since mathematical non-selfadjointness is reversible via a mathematical transformation, we refer to it as pseudo non-selfadjointness, while physical non-selfadjointness, which cannot be removed mathematically, is called simply non-selfadjointness. Although it is possible to find a transformation that will convert the apparent non-selfadjointness back to its original selfadjoint state, the effort and time involved in looking for one may actually overwhelm the slightly more tedious task of solving it as a non-selfadjoint problem in the first instance. Therefore, we have proposed a systematic approach to treat non-selfadjoint problems in structural dynamics. It includes extending existing procedures for differential operators to cope with differential matrix operators which are commonly encountered in structural dynamics. For efficient handling, a direct technique to obtain the formal adjoint of such an operator is provided. The adjoint analysis is then completed by deriving the boundary conditions defining the manifolds of the original and adjoint systems. It is shown that for a free vibration analysis of a non-selfadjoint system, only the eigenpairs comprising the eigenvalues and eigenvectors of the original systems are required. On the other hand, for a forced vibration analysis, it is necessary to determine the eigenpairs of both the original and the adjoint systems. To relate the eigenvectors of these two systems for the subsequent response calculations, the biorthogonality conditions can be applied. It is hoped that the method introduced here will provide an alternative and powerful tool for solving general dynamic problems.