Perturbation of invariant subspaces

We consider two different theoretical approaches for the problem of the perturbation of invariant subspaces. The first approach belongs to the standard theory. In that approach the bounds for the norm of the perturbation of the projector are proportional to the norm of perturbation matrix, and inversely proportional to the distance between the corresponding eigenvalues and the rest of the spectrum. The second approach belongs to the relative theory which deals only with Hermitian matrices. The bounds which result from this approach are proportional to the size of relative perturbation of matrix elements and the condition number of a scaled matrix, and inversely proportional to the relative gap between the corresponding eigenvalue and the rest of the spectrum. Because of a relative gap these bounds are in some cases less pessimistic than the standard norm estimates

On some properties of the Lyapunov equation for damped systems

We consider a damped linear vibrational system whose dampers depend linearly on the viscosity parameter v. We show that the trace of the corresponding Lyapunov solution can be represented as a rational function of v whose poles are the eigenvalues of a certain skew symmetric matrix. This makes it possible to derive an asymptotic expansion of the solution in the neighborhood of zero (small damping)

Relative perturbation of invariant subspaces

In this paper we consider the upper bound for the sine of the greatest canonical angle between the original invariant subspace and its perturbation. We present our recent results which generalize some of the results from the relative perturbation theory of indefinite Hermitian matrices