Perturbation analysis of Lagrangian invariant subspaces of symplectic matrices

Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle. We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability as well as the index of stability for these subspaces.