On the Congruent Selection of Jordan Blocks from a Singular Square Matrix

Abstract

The concept of a regularizing decomposition was introduced by R. Horn and V. Sergeichuk. It means the representation of a square matrix by a direct sum of Jordan blocks with zero on the principal diagonal and a nonsingular matrix. Such a representation is attained via congruence transformations and differs from the Jordan normal form. For the reasons explained in this paper, we prefer to speak of an SR decomposition (in other words, a singular-regular decomposition) of a matrix rather than a regularizing decomposition. Accordingly, algorithms providing this decomposition are called SR algorithms.We develop a rational algorithm that considerably simplifies the SR algorithms proposed by Horn and Sergeichuk.