Geodesic
On congruent selection of the Jordan blocks from a~singular square matrix
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 3, pp. 255-258.

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The concept of a regularizing decomposition was introduced by R. Horn and V. Sergeichuk. This means the representation of a square matrix by a direct sum of the Jordan blocks with zero on the principal diagonal and a non-singular matrix. Such a representation is attained via congruent transformations and differs from the Jordan normal form. For the reasons explained in this paper, we prefer to speak about the SN-decomposition of a matrix (in other words, singular-non-singular decomposition) rather than the regularizing decomposition. Accordingly, the algorithms providing the former decomposition are called SN-algorithms. We propose a rational algorithm that considerably simplifies the SN-algorithms proposed by Horn and Sergeichuk. 
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     title = {On congruent selection of the {Jordan} blocks from a~singular square matrix},
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Kh. D. Ikramov. On congruent selection of the Jordan blocks from a~singular square matrix. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 3, pp. 255-258. http://geodesic.mathdoc.fr/item/SJVM_2018_21_3_a1/

[1] Horn R. A., Sergeichuk V. V., “A regularization algorithm for matrices of bilinear and sesquilinear forms”, Linear Algebra Appl., 412:2–3 (2006), 380–395 | DOI | MR | Zbl