Matrix linear unilateral and bilateral equations over quadratic rings

Ladzoryshyn, N. B.; Petrychkovych, V. М.

doi:10.30970/vmm.2018.85.032-040

Cited by 4 publications

(2 citation statements)

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“…In [2], authors use obtained in [16,17] the standard form of pair of matrices relative to generalized equivalence for describing solutions of matrix unilateral and bilateral equations over the principal ideal rings and Bezout rings. In [11], the criteria for solvability of matrix linear unilateral and bilateral equations over arbitrary quadratic rings are established, and in [9] integer solutions of these equations are described. In this article, it is proposed a method of constructing solutions of the matrix Diophantine equations over quadratic rings and described their structure with application.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

“…The Euclidean norm E (H) of the matrix H from the solution (11) of the matrix equation (9) is less than the Euclidean norm E (µ B n ) of the last invariant multiplier µ B n of the matrix H from the equation (5). The Euclidean norm E (D B ) of the canonical diagonal form D B of the matrix B equals to the Euclidean norm E (µ B n ) of the invariant multiplier µ B n .…”

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confidence: 99%

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Matrix Diophantine equations over quadratic rings and their solutions

Ladzoryshyn

Petrychkovych

Zelisko

2020

Carpathian Math. Publ.

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The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

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