Матричні лінійні різносторонні рівняння над різними областями, методи побудови розв’язків та опис їхньої структури

N. S. Dzhaliuk, V. M. Petrychkovych

Анотація


Наведено огляд методів розв’язування матричних лінійних різносторонніх рівнянь, зокрема рівнянь типу Сильвестра над різними областями та опису структури їхніх розв’язків. Основну увагу зосереджено на розширенні і узагальненні результатів, одержаних авторами раніше. На основі стандартної форми поліноміальних матриць відносно напівскалярної еквівалентності розроблено метод розв’язування матричних поліноміальних рівнянь типу Сильвестра. Досліджено структуру їхніх розв’язків. Виділено розв’язки обмежених степенів і наведено умови єдиності цих розв’язків. Запропоновано метод побудови розв’язків матричних рівнянь Сильвестра над адекватними кільцями, а також встановлено критерії єдиності розв’язків певного вигляду. Встановлено умови існування розв’язку матричного рівняння Сильвестра укільцях трикутних та блочно-трикутних матриць над комутативною областю головних ідеалів.

 

Зразок для цитування: Н. С. Джалюк, В. М. Петричкович, “Матричні лінійні різносторонні рівняння над різними областями, методи побудови розв’язків та опис їхньої структури,” Мат. методи та фіз.-мех. поля, 65, No. 1-2, 18–41 (2022), https://doi.org/10.15407/mmpmf2022.65.1-2.18-41


Ключові слова


поліноміальне кільце, адекватне кільце, матриця, еквівалент- ність, напівскалярна еквівалентність, узагальнена еквівалентність, матричне рівняння

Посилання


F. A. Aliev, V. B. Larin, “Special cases of optimization problems for stationary linear closed-loop systems,” Prikl. Mekh., 39, No. 3, 3-26 (2003); English translation: Int. Appl. Mech., 39, No. 3, 251-273 (2003), https://doi.org/10.1023/A:1024433417982

N. S. Dzhaliuk, “Existence of the solution of the Sylvester-type matrix equation in the ring of block triangular matrices,” Prykl. Probl. Mekh. Mat., Issue 19, 79–83 (2021) (in Ukrainian), https://doi.org/10.15407/apmm2021.19.79-83

N. S. Dzhaliuk, “The uniqueness of the block-triangular factorizations of matrices over principal ideal rings,” Dop. Nats. Akad. Nauk Ukr., Ser. Mat. Pryr. Tekhn. Nauky, 2010, No. 1, 7–12 (2010) (in Ukrainian).

N. S. Dzhaliuk, “Solutions of the matrix equation $AX+YB=C$ with triangular coefficients,” Mat. Met. Fiz.-Mekh. Polya, 62, No. 2, 26−31 (2019); English translation: J. Math. Sci., 261, No. 1, 25–32 (2022), https://doi.org/10.1007/s10958-022-05734-x

N. S. Dzhaliuk, V. M. Petrychkovych, “Equivalence of matrices in the ring $M(n,R)$ and its subrings,” Ukr. Mat. Zh., 73, No. 12, 1612–1618 (2021); English translation: Ukr. Math. J. 73, No. 12, 1865-1872 (2022); https://doi.org/10.1007/s11253-022-02034-0

N. Dzhaliuk, V. Petrychkovych, “The semiscalar equivalence of polynomial matrices and the solution of the Sylvester matrix polynomial equations,” Mat. Visn. NTSh, 9, 81–88 (2012) (in Ukrainian).

V. R. Zelisko, N. B. Ladzoryshyn, V. M. Petrychkovych, “On equivalence of the matrices over quadratic Euclidian rings,” Prykl. Probl. Mekh. Mat., Issue 4, 16–21 (2006).

P. S. Kazimirs’kyi, Factorization of Matrix Polynomials [in Ukrainian], IAPMM, NAS of Ukraine, Lviv (2015).

P. S. Kazimirs’kyi, V. M. Petrychkovych, “On equivalence of polynomial matrices,” in: Theoretical and Applied Problems of Algebra and Differential Equations [in Ukrainian], Nauk. Dumka, Kyiv (1977), pp. 61–66.

D. G. Korenevskij, “Stability of the solutions of deterministic and stochastic differential-difference equations (algebraic criteria),” Nauk. Dumka, Kyiv (1992) (in Russian).

N. B. Ladzoryshyn, “On equivalence of pairs of matrices, which determinants are primes powers, over quadratic Euclidean rings,” Carpathian Math. Publ., 5 (1), 63–69 (2013) (in Ukrainian); https://doi.org/10.15330/cmp.5.1.63-69.

N. B. Ladzoryshyn, “Integer solutions of matrix linear unilateral and bilateral equations over quadratic rings,” Mat. Met. Fiz.-Mekh. Polya, 58, No. 2, 47−54 (2015); English translation: J. Math. Sci., 223, No. 1, 50–59 (2017), https://doi.org/10.1007/s10958-017-3337-0

N. B. Ladzoryshyn, 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), https://doi.org/10.1007/s10958-021-05212-w

A. G. Mazko, Robust Stability and Stabilization of Dynamic Systems: Methods of Matrix and Cone Inequalities [in Russian], Vol. 102 of Mathematics and Its Application: Proceedings of the Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (2016).

V. M. Petrichkovich, “Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices,” Mat. Zametki, 37, No. 6, 789–796 (1985); English translation: Math. Notes, 37, No. 6, 431–435 (1985), https://doi.org/10.1007/BF01157677

V. M. Petrichkovich, “Semiscalar equivalence and the Smith normal form of polynomial matrices,” Mat. Met. Fiz.-Mekh. Polya, Iss. 26, 13−16 (1987) (in Russian); English translation: J. Sov. Math., 66, No. 1, 2030–2033 (1993), https://doi.org/10.1007/BF01097386

V. M. Petrichkovich, “Semiscalar equivalence and the factorization of polynomial matrices,” Ukr. Mat. Zh., 42, No. 5, 644–649 (1990); English translation: Ukr. Math. J., 42, No. 5, 570–574 (1990), https://doi.org/10.1007/BF01065057

V. M. Petrychkovych, “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,” Mat. Met. Fiz.-Mekh. Polya, 62, No. 4, 7−27 (2018); English translation: J. Math. Sci., 265, No. 2, 345–368 (2022), https://doi.org/10.1007/s10958-022-06057-7

V. M. Petrychkovych, Generalized Equivalence of Matrices and Its Collections and Factorization of Matrices over Rings [in Ukrainian], IAPMM, NAS of Ukraine, Lviv (2015).

H. Abou-Kandil, G. Freiling, V. Ionescu, G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser–Springer (2003), https://doi.org/10.1007/978-3-0348-8081-7

S. Barnett, Matrices in Control Theory with Applications to Linear Programming, Van Nostrand Reinhold, London (1971).

S. Barnett, “Regular polynomial matrices having relatively prime determinants,” Math. Proc. Cambridge Philos. Soc., 65, No. 3, 585–590 (1969), https://doi.org/10.1017/S0305004100003364

H. Bart, A. P. M. Wagelmans, “An integer programming problem and rank decomposition of block upper triangular matrices,” Linear Algebra Appl., 305, No. 1-3, 107–129 (2000), https://doi.org/10.1016/S0024-3795(99)00219-0

S. Chen, Y. Tian, “On solutions of generalized Sylvester equation in polynomial matrices,” J. Franklin Inst., 351, No. 12, 5376–5385 (2014), http://doi.org/10.1016/j.jfranklin.2014.09.024

J. A. Dias da Silva, T. J. Laffey, “On simultaneous similarity of matrices and related questions,” Linear Algebra Appl., 291, No. 1-3, 167–184 (1999), https://doi.org/10.1016/S0024-3795(98)10247-1

A. Dmytryshyn, V. Futorny, T. Klymchuk, V. V. Sergeichuk, “Generalization of Roth’s solvability criteria to systems of matrix equations,” Linear Algebra Appl., 527, 294–302 (2017), https://doi.org/10.1016/j.laa.2017.04.011

A. Dmytryshyn, B. Kågström, “Coupled Sylvester-type matrix equations and block diagonalization,” SIAM J. Matrix Anal. Appl., 36, No. 2, 580–593 (2015), https://doi.org/10.1137/151005907

N. S. Dzhaliuk, V. M. Petrychkovych, “Solutions of the matrix linear bilateral polynomial equation and their structure,” Algebra Discrete Math., 27, No. 2, 243–251 (2019).

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

R. B. Feinberg, “Equivalence of partitioned matrices,” J. Res. Natl. Bur. Stand., 80B, No. 1, 89–97 (1976), https://doi.org/10.6028/jres.080B.015

J. Feinstein, Y. Bar-Ness, “On the uniqueness of the minimal solution to the matrix polynomial equation $A(λ)X(λ) + Y(λ)B(λ)=C(λ)$,” J. Franklin Inst., 310, No. 2, 131–134 (1980), https://doi.org/10.1016/0016-0032(78)90012-1

V. Futorny, T. Klymchuk, V. V. Sergeichuk, “Roth’s solvability criteria for the matrix equations $AX$-hat{$X$}$B=C$ and $X-A$hat{$X$}$B=C$ over the skew field of quaternions with an involutive automorphism $q$→hat{$q$},” Linear Algebra Appl., 510, 246–258 (2016), https://doi.org/10.1016/j.laa.2016.08.022

W. H. Gustafson, “Roth’s theorems over commutative rings,” Linear Algebra Appl., 23, 245–251 (1979), https://doi.org/10.1016/0024-3795(79)90106-X

R. E. Hartwig, “Roth’s equivalence problem in unit regular rings,” Proc. Amer. Math. Soc., 59, 39–44 (1976), https://doi.org/10.1090/S0002-9939-1976-0409543-4

R. E. Hartwig, P. Patricio, “On Roth’s pseudo equivalence over rings,” Electron. J. Linear Algebra, 16, 111–124 (2007), https://doi.org/10.13001/1081-3810.1187

Zhuo-Heng He, “Pure PSVD approach to Sylvester-type quaternion matrix equations,” Electron. J. Linear Algebra, 35, 266–284 (2019), https://doi.org/10.13001/1081-3810.3917

O. Helmer, “The elementary divisor theorem for certain rings without chain condition,” Bull. Amer. Math. Soc., 49, No. 4, 225–236 (1943), https://doi.org/10.1090/S0002-9904-1943-07886-X

L. Huang, J. Liu, “The extention of Roth’s theorem for matrix equations over a ring,” Linear Algebra Appl., 259, 229–235 (1997), https://doi.org/10.1016/S0024-3795(96)00286-8

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

T. Kaczorek, Polynomial and Rational Matrices: Applications in Dynamical Systems Theory, Springer, London (2007), https://doi.org/10.1007/978-1-84628-605-6

T. Kaczorek, “Zero-degree solutions to the bilateral polynomial matrix equations,” Bull. Polish Acad. Sci. Ser. Techn. Sci., 34, No. 9–10, 547–552 (1986).

I. Kaplansky, “Elementary divisors and modules,” Trans. Amer. Math. Soc., 66, 464–491 (1949), https://doi.org/10.1090/S0002-9947-1949-0031470-3

V. Kučera, “Algebraic theory of discrete optimal control for multivariable systems,” Kybernetika, 10, No. 7, 1–56 (1974).

V. Kučera, “Algebraic theory of discrete optimal control for single-variable systems. I. Preliminaries,” Kybernetika, 9, No. 2, 94–107 (1973).

N. B. Ladzoryshyn, V. M. Petrychkovych, “The number of standard forms of matrices over imaginary Euclidean quadratic rings with respect to the $(z,k)$-equivalence,” Mat. Stud., 57, No. 2, 115–121 (2022), https://doi.org/10.30970/ms.57.2.115-121

N. B. Ladzoryshyn, V. M. Petrychkovych, H. V. Zelisko, “Matrix Diophantine equations over quadratic rings and their solutions,” Carpathian Math. Publ., 12, No. 2, 368-375 (2020), https://doi.org/10.15330/cmp.12.2.368-375

N. Ladzoryshyn, V. Petrychkovych, “Equivalence of pairs of matrices with relatively prime determinants over quadratic rings of principal ideals,” Bul. Acad. Ştiinţe Repub. Mold. Mat., No. 3(76), 38–48 (2014).

P. Lancaster, L. Rodman, Algebraic Riccati equations, Clarendon Press, Oxford (1995).

Liu Yong Hui, “Ranks of solutions of the linear matrix equation $AX+YB=C$,” Comput. Math. Appl., 52, No. 6-7, 861–872 (2006), https://doi.org/10.1016/j.camwa.2006.05.011

F. Martins, E. Pereira, “Block matrices and stability theory,” Tatra Mt. Math. Publ., 38, No. 4, 147–162 (2007).

M. Newman, “The Smith normal form of a partitioned matrix,” J. Res. Natl. Bur. Stand., 78B, No. 1, 3–6 (1974), https://doi.org/10.6028/JRES.078B.002

A. B. Özgüler, “The equation $AXB+CYD=E$ over a principal ideal domain,” SIAM J. Matrix Anal. Appl., 12, No. 3, 581–591 (1991), https://doi.org/10.1137/0612044

V. Petrychkovych, “Generalized equivalence of pairs of matrices,” Linear Multilinear Algebra, 48, No. 2, 179–188 (2000), https://doi.org/10.1080/03081080008818667

V. Petrychkovych, “Standard form of pairs of matrices with respect to generalized equivalence,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Iss. 61, 148–155 (2003).

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

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

V. Simoncini, “Computational methods for linear matrix equations,” SIAM Review, 58, No. 3, 377–441 (2016), https://doi.org/10.1137/130912839

J. J. Sylvester, “Sur les racines des matrices unitaires,” Compt. Rend., 94, 396–399 (1882).

Y. Tian, “Completing triangular block matrices with maximal and minimal ranks,” Linear Algebra Appl., 321, No. 1-3, 327–345 (2000), https://doi.org/10.1016/S0024-3795(00)00224-X

Y. Tian, “Upper and lower bounds for ranks of matrix expressions using generalized inverses,” Linear Algebra Appl., 355, No. 1-3, 187–214 (2002), https://doi.org/10.1016/S0024-3795(02)00345-2

Y. Tian, C. Xia, “On the low-degree solution of the Sylvester matrix polynomial equation,” Hindawi J. Math., 2021, Article ID 4612177 (2021), https://doi.org/10.1155/2021/4612177

P. A. Tzekis, “A new algorithm for the solution of a polynomial matrix Diophantine equation,” Appl. Math. Comput., 193, No. 2, 395–407 (2007), https://doi.org/10.1016/j.amc.2007.03.076

W. A. Wolovich, P. J. Antsaklis, “The canonical Diophantine equations with applications,” SIAM J. Control Optim., 22, No. 5, 777–787 (1984), https://doi.org/10.1137/0322049

B. Zhou, Z.-B. Yan, G.-R. Duan, “Unified parametrization for the solutions to the polynomial Diophantine matrix equation and the generalized Sylvester matrix equation,” Int. J. Control Autom. Syst., 8, No. 1, 29–35 (2010), https://doi.org/10.1007/s12555-010-0104-0


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