Skip to main content
SpringerLink
Log in

Solutions of the Matrix Equation AX + YB = C with Triangular Coefficients

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We establish necessary and sufficient conditions for the existence of triangular solutions of a linear matrix equation AX + YB = Cover the commutative ring of principal ideals whose matrix coefficients A, B , and C are triangular matrices. It is also shown that there is no matrix equation of this kind all solutions of which are triangular.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. B. Ladzoryshyn and V. M. Petrychkovych, “Standard form of matrices over quadratic rings with respect to the (z,k)-equivalence and the structure of solutions of bilateral matrix linear equations,” Mat. Met. Fiz.-Mekh. Polya, 61, No. 2, 49–56 (2018); English translation: J. Math. Sci., 253, No. 1, 54–62 (2021).

  2. N. S. Dzhaliuk and V. M. Petrychkovych, “The matrix linear unilateral and bilateral equations with two variables over commutative rings,” Int. Scholarly Res. Notices. ISRN Algebra, 2012, Article ID 205478 (2012); https://doi.org/10.5402/2012/205478.

    MathSciNet  MATH  Google Scholar 

  3. R. B. Feinberg, “Equivalence of partitioned matrices,” J. Res. Nat. Bul. Stand., 80B, No. 1, 89–97 (1976).

    MathSciNet  MATH  Google Scholar 

  4. I. Jonsson and B. Kågström, “Recursive blocked algorithms for solving triangular systems. Part I: One-sided and coupled Sylvestertype matrix equations,” ACM Trans. Math. Software, 28, No. 4, 392–415 (2002); https://doi.org/10.1145/592843.592845.

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Kaczorek, Polynomial and Rational Matrices. Applications in Dynamical Systems Theory, Springer, London (2007).

    Book  Google Scholar 

  6. V. Petrychkovych and N. Dzhaliuk, “Factorizations in the rings of the block matrices,” Bul. Acad. Ştiinţe Repub. Mold. Mat., No. 3 (85), 23–33 (2017).

    MathSciNet  MATH  Google Scholar 

  7. W. E. Roth, “The equations AX − YB = C and AX − XB = C in matrices,” Proc. Am. Math. Soc., 3, No. 3, 392–396 (1952); https://doi.org/10.2307/2031890.

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Tian, “Completing triangular block matrices with maximal and minimal ranks,” Linear Alg. Appl., 321, Nos. 1-3, 327–345 (2000); https://doi.org/10.1016/S0024-3795(00)00224-X.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    N. S. Dzhaliuk

Authors
  1. N. S. Dzhaliuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. S. Dzhaliuk.

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 62, No. 2, pp. 26–31, April–June, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dzhaliuk, N.S. Solutions of the Matrix Equation AX + YB = C with Triangular Coefficients. J Math Sci 261, 25–32 (2022). https://doi.org/10.1007/s10958-022-05734-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-05734-x

Keywords

Search

Navigation