On Some Bounds on the Perturbation of Invariant Subspaces of Normal Matrices with Application to a Graph Connection Problem

We provide upper bounds on the perturbation of invariant subspaces of normal matrices measured using a metric on the space of vector subspaces of $\mathbb{C}^n$ in terms of the spectrum of both the unperturbed \& perturbed matrices, as well as, spectrum of the unperturbed matrix only. The results presented give tighter bounds than the Davis-Khan $\sin\Theta$ theorem. We apply the result to a graph perturbation problem.