On the noncongruence of direct sums of nilpotent Jordan blocks

Recieved: 01/09/2024

Accepted: 06/12/2024

Published: 01/31/2025

Keywords: direct sum, Jordan block, congruence transformation, span of a system of vectors

To cite this article

Ikramov Kh.D. On the noncongruence of direct sums of nilpotent Jordan blocks. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2025. N 1, p.49-56 https://doi.org/10.55959/MSU/0137–0782–15–2025–49–1–49–56.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Ikramov Kh.D. (Lomonosov Moscow State University (MSU))

Abstract

Let A and B be matrices of order n that are direct sums of nilpotent Jordan blocks. Suppose that A and B are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, A and B cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.