Distance to Singularity for Skew-Symmetric Matrix Pencils

Abstract

The singularity of a matrix or matrix pencil is easily affected by small errors in numerical calculations. Therefore, it is motivated to consider the distance to singularity, rather than whether or not the matrix (pencil) is singular. In the case of unstructured matrices, the distance to singularity is well known. For structured matrices and matrix pencils, however, the question is more complex. In this thesis, a numerical method for determining the distance to the nearest skew-symmetric matrix pencil of given maximal rank is presented. The task is formulated as a minimization problem using the skew-symmetric Kronecker Canonical form and a rank-1 decomposition of skew-symmetric matrix pencils. Four different algorithms for solving the minimization problem are proposed, using the so-called vec-trick, QRdecomposition, singular value decomposition, and the GUPTRI form [5, 6]. These algorithms perform well compared to state of the art methods.