A Method for Computing the Distance of a Stable Matrix to the Set of Unstable Matrices.

Abstract

We propose a method to compute the spectral norm distance from a given matrix A to the set of matrices having at least an eigenvalue on the imaginary axis. It is shown that the distance is one of the roots of a suitably constructed polynomial in one variable. Our method can be easily generalized to compute the distance from A to the set of matrices having at least an eigenvalue on any straight line or circle. Thus, it can be applied to compute the distance from a stable matrix to the set of unstable matrices in either continuous or discrete sense.