### Abstract:

The primary aim of this thesis is to develop a framework for direct methods for solutions of rational eigenvalue problems. To achieve this goal, we propose to reformulate the problem of solving a rational eigenvalue problem G( )u = 0 for 2 C and nonzero vector u 2 Cn to that of computation of transmission zeros and zero directions of a linear time invariant (LTI) system given by Ex (t) = Ax(t) + Bu(t) y(t) = Cx(t) + P( d dt )u(t) for which G( ) is the transfer function. Then the eigenvalues of G( ) form a subset of the transmission zeros of the LTI system . On the other hand, transmission zeros of are subset of the invariant zeros of the system and the invariant zeros are eigenvalues of the Rosenbrock system matrix S( ) given by S( ) = 2 4 P( ) C B (A E) 3 5 . The transmission zeros of coincide with the invariant zeros when the LTI system is controllable as well as observable. We therefore develop a framework for computing eigenvalues and eigenvectors of the the Rosenbrock system matrix S( ). For this purpose, we introduce three families of linearizations - which we refer to as Fiedler pencils, Generalized Fiedler (GF) pencils and Generalized Fiedler pencils with repetition (GFPR) - of the Rosenbrock system matrix S( ). We solve the eigenvalue problem for these Fiedler pencils to obtain invariant zeros of the LTI system . Thus, schematically, our strategy for numerical solution of rational eigenvalue problem is as follows: Rational matrix function State-space realization Linearization Solution. There are efficient methods for computing a (minimal) state-space realization of a rational matrix function. We therefore focus on the construction of Fiedler pencils of the Rosenbrock system matrix S( ) and show that these Fiedler pencils are linearizations for S( ). Further, we show that the Fiedler pencils of S( ) are linearizations of G( ) when the LTI system is both controllable and observable, that is, when is a minimal state-space realization of the transfer function G( ). We also describe eigenvector recovery of G( ) from that of Fiedler pencils of...