Home – Home – Jornals – Numerical Analysis and Applications 2026 number 3

Suppose one needs to check whether given square matrices A and B are congruent. If A and B are nonsingular, one can use such useful instruments as the cosquares of both matrices. How can one reduce the case of singular A and B to a nonsingular one? Sergeichuk and Horn in 2006 proposed a method that brings a given singular matrix A by a finite sequence of congruences to a direct sum of a nonsingular submatrix and one or several nilpotent Jordan blocks. The blocks constitute the singular part of the canonical form of A with respect to congruences. Removing them from A yields the desired method that replaces the input singular matrix by a nonsingular submatrix of smaller order. The method of Horn-Sergeichuk was not implemented by its authors as a working procedure. In this paper, we describe our implementation of this method in the form of an algorithm for rational matrices or complex matrices whose entries are rational Gaussian numbers. The algorithm is based on the error-free arithmetic of rational numbers, which is provided to the users of the computational environment Matlab. The performance of the algorithm is illustrated by two numerical examples.