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Standard Forms of the Matrices Over Rings with Respect to Various Types of Equivalence and Their Applications to the Theory of Matrix Factorization and Matrix Equations

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We present a survey of the results of investigations of one of the fields dealing with the equivalence of matrices originated by P. S. Kazimirs’kii and then continued and developed by his colleagues. We formulate standard forms of polynomial matrices and their finite sets with respect to the semiscalar equivalence and generalized equivalence of the pairs of matrices over rings. Their applications to the development of the methods of matrix factorization, solving of the matrix equations, description of the structure of solutions of these equations and, in particular, of the Sylvester matrix equations, linear Diophantine matrix equations, and other problems are indicated.

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  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 62, No. 4, pp. 7–27, October–December, 2019.

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Petrychkovych, V.M. Standard Forms of the Matrices Over Rings with Respect to Various Types of Equivalence and Their Applications to the Theory of Matrix Factorization and Matrix Equations. J Math Sci 265, 345–368 (2022). https://doi.org/10.1007/s10958-022-06057-7

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