Canonical form of low-dimensional matrices with respect to one-sided transformations

V.P. Shchedryk

doi:10.15421/242610

Canonical form of low-dimensional matrices with respect to one-sided transformations

V.P. Shchedryk (Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine), https://orcid.org/0000-0003-4536-5276

Abstract

We construct a new canonical form for third-order matrices with respect to one-sided transformations. Unlike the classical triangular Hermite form, our canonical form is expressed as the product of two matrices: $$$P\Phi$$$, where $$$P$$$ is invertible and $$$\Phi$$$ is in Smith form. This form enables the description of all non-associated matrices with a given Smith form, which, in particular, allows us to find all roots of matrix polynomial equations and to parametrically describe the third-order general linear group over a field.

Keywords

one-sided transformation; canonical form; Zelisko group; $$$\Phi$$$-skeleton

MSC 2020

15A21; 15A23; 16U30

Full Text:

PDF

References

Bovdi V.A., Shchedryk V.P. "Generating solutions of a linear equation and structure of elements of the Zelisko group", Linear Algebra Applic., 2021; 625(1): pp. 55-67. doi:10.1016/j.laa.2021.04.019

Bovdi V.A., Shchedryk V.P. "Generating solutions of a linear equation and structure of elements of the Zelisko group II", Quaest. Math., 2023; 46(9): pp. 1789-1798. doi:10.2989/16073606.2022.2112629

Helmer O. "The elementary divisor theorem for certain rings without chain condition", Bull. Amer. Math. Soc., 1943; 49(4): pp. 225-236. doi:10.1090/S0002-9904-1943-07886-X

Hermite C. "Sur l'introduction des variables continues dans la théorie des nombres", J. Reine Angew. Math., 1851; 41(1): pp. 191-216.

Kaplansky I. "Elementary divisors and modules", Trans. Amer. Math. Soc., 1949; 66(2): pp. 464-491. doi:10.2307/1990591

Kazimirskii P.S. Factorization of Matrix Polynomials, Naukova Dumka, 1981.

Kazimirskii P.S. "Solving the problem of separating a regular factor from a matrix polynomial", Ukrainian Math. J., 1980; 32(4): pp. 483-498.

Shchedryk V.P. Arithmetic of Matrices over Rings, Akademperiodyka, 2021. doi:10.15407/akademperiodika.430.278

Shchedryk V.P. "Parametric 2-decompositions in complete linear groups of small order over a field", Ukrainian Math. J., 2024; 76(2): pp. 330-348. doi:10.1007/s11253-024-02323-w

Shchedryk V.P. "On transforming matrices and matrix divisors over some Bézout domains", Math. Methods Physicomechan. Fields, 2000; 43(1): pp. 36-44.

Shchedryk V.P. "Non-associated matrices with a standard $$$\Phi$$$-skeleton", Math. Methods Physicomechan. Fields, 2002; 45(3): pp. 32-44.

Shchedryk V.P. "Some determinant properties of primitive matrices over Bezout B-domain", Algebra Discrete Math., 2005; 2: pp. 46-57.

Shchedryk V.P. "Bezout rings of stable range 1.5", Ukrainian Math. J., 2015; 67(6): pp. 960-974. doi:10.1007/s11253-015-1126-9

Smith H.J.S. "On systems of linear indeterminate equations and congruences", Philos. Trans. Royal Soc. London, 1861; 151(1): pp. 293-326.

Zelisko V.R. "On the structure of some class of invertible matrices", Math. Methods Physicomechan. Fields, 1980; 12(1): pp. 14-21.