This dissertation concerns the development of a perturbation theory applicable to the algebraic eigenvalue problem. The objective is to express the perturbed eigenvalues and eigenvectors in terms of the unperturbed eigenvalues and eigenvectors and the perturbing matrices, without solving a new eigenvalue problem. If all of the unperturbed eigenvalues are clearly distinct, which is to say that the difference between no two of them is small, this objective can be accomplished. Alternatively, if some of the unperturbed eigenvalues are not clearly distinct, an eigenvalue problem, of the same dimension as the number of not clearly distinct unperturbed eigenvalues, must be solved. It is shown that Rayleigh's quotient is related to the perturbation theory. The use of Rayleigh's quotient is a key ingredient in defining a generalized basis for application of the perturbation theory, which does not depend upon the use of eigenvectors satisfying an eigenvalue problem. This generalized basis is then used to define an iterative eigensolution algorithm. The perturbation theory was initially developed as an efficient means of solving the eigenvalue problem associated with linear gyroscopic systems for which the damping and/or circulatory effects are sufficiently small that they can be regarded as perturbations. The perturbation theory is applied to an example of such a system. One interesting aspect of this example is that for one of the combinations of parameters, it exhibits both divergence and flutter instability, simultaneously. Other numerical examples are given. / Ph. D.