On the Nearest Singular Matrix Pencil
Nicola Guglielmi, Christian Lubich, Volker Mehrmann · SIAM Journal on Matrix Analysis and Applications · 2017
Given a regular matrix pencil $A + \mu E$, we consider the problem of determining the nearest singular matrix pencil with respect to the Frobenius norm. We present new approaches based on the solution of matrix differential equations for determining the nearest singular pencil $A + \Delta A +\mu( E + \Delta E)$: one approach for general singular pencils and another one such that $A+\Delta A$ and $E+\Delta E$ have a common left/right null vector. For the latter case the nearest singular pencil is shown to differ from the original pencil by rank-one matrices $\Delta A$ and $\Delta E$. In both cases we consider also the situation where only $A$ is perturbed. The nearest singular pencil is approached by a two-level iteration, where a gradient flow is driven to an equilibrium point in the inner iteration and the outer level uses a fast iteration for the distance parameter. This approach extends also to structured matrices $A$ and $E$.