Solution of a Linear Nondegenerate Matrix Equation Based on the Zero Divisor

Авторы: Zubov N.E., Ryabchenko V.N. Опубликовано: 03.11.2021
Опубликовано в выпуске: #5(98)/2021  
DOI: 10.18698/1812-3368-2021-5-49-59

Раздел: Математика | Рубрика: Дифференциальные уравнения, динамические системы и оптимальное управление  
Ключевые слова: linear nondegenerate matrix equation, determinant calculation, zero divisor, solution formula

New formulas were obtained to solve the linear non-degenerate matrix equations based on zero divisors of numerical matrices. Two theorems were formulated, and a proof to one of them is provided. It is noted that the proof of the second theorem is similar to the proof of the first one. The proved theorem substantiates new formula in solving the equation equivalent in the sense of the solution uniqueness to the known formulas. Its fundamental difference lies in the following: any explicit matrix inversion or determinant calculation is missing; solution is "based" not on the left, but on the right side of the matrix equation; zero divisor method is used (it was never used in classical formulas for solving a matrix equation); zero divisor calculation is reduced to simple operations of permutating the vector elements on the right-hand side of the matrix equation. Examples are provided of applying the proposed method for solving a nondegenerate matrix equation to the numerical matrix equations. High accuracy of the proposed formulas for solving the matrix equations is demonstrated in comparison with standard solvers used in the MATLAB environment. Similar problems arise in the synthesis of fast and ultrafast iterative solvers of linear matrix equations, as well as in nonparametric identification of abnormal (emergency) modes in complex technical systems, for example, in the power systems


[1] Voevodin V.V., Kuznetsov Yu.A. Matritsy i vychisleniya [Matrices and calculations]. Moscow, Nauka Publ., 1984.

[2] Gantmakher F.R. Teoriya matrits [Matrix theory]. Moscow, Nauka Publ., 1966.

[3] Golub G.H., Van Loan C.F. Matrix computations. Johns Hopkins Univ. Press, 1996.

[4] Il’in V.A., Poznyak E.G. Lineynaya algebra [Linear algebra]. Moscow, Nauka Publ., 1984.

[5] Il’in V.P. Metody nepolnoy faktorizatsii dlya resheniya algebraicheskikh system [Incomplete factorization methods for solving algebraic systems]. Moscow, FIZMATLIT Publ., 1995.

[6] Taufer J. Losung der Randwertprobleme fur Systeme von Linearen Differentialgleichungen. Praha, Academia, 1973.

[7] Kailath T. Linear systems. Prentice Hall, 1980.

[8] Bernstein D.S. Matrix mathematics. Princeton Univ. Press, 2005.

[9] Zubov N.E., Mikrin E.A., Ryabchenko V.N. Matrichnye metody v teorii i praktike sistem avtomaticheskogo upravleniya letatel’nykh apparatov [Matrix methods in theory and practice of automatic control systems of aircraft]. Moscow, BMSTU Publ., 2016.

[10] Helmke U., Jordan J. Control and stabilization of linear equation solvers. In: Willems J.C., Hara S., Ohta Y., Fujioka H. (eds). Perspectives in Mathematical System Theory, Control, and Signal Processing. Lecture Notes in Control and Information Sciences, vol. 398. Berlin, Heidelberg, Springer, 2010, pp. 73--82. DOI: https://doi.org/10.1007/978-3-540-93918-4_7

[11] Helmke U., Jordan J. Optimal control of iterative solution methods for linear systems of equations. Proc. Appl. Math. Mech., 2005, vol. 5, iss. 1, pp. 163--164.DOI: https://doi.org/10.1002/pamm.200510061

[12] Helmke U., Jordan J. (eds). Mathematical systems theory in biology, communications, computations and finance. The IMA Volumes in Mathematics and its Applications, vol. 134. New York, NY, Springer, 2002. DOI: https://doi.org/10.1007/978-0-387-21696-6_1

[13] Gadzhiev M.G., Zhgun K.V., Zubov N.E., et al. Synthesis of fast and superfast solvers of large systems of linear algebraic equations using control theory methods. J. Comput. Syst. Sci. Int., 2019, vol. 58, no. 4, pp. 560--570. DOI: https://doi.org/10.1134/S1064230719020084

[14] Mikrin E.A., Zubov N.E., Efanov D.V., et al. Superfast iterative solvers for linear matrix equations. Doklady Mathematics, 2018, vol. 98, no. 2, pp. 444--447. DOI: https://doi.org/10.1134/S1064562418060145

[15] Galiaskarov I.M., Zubov N.E., Zybin E.Yu., et al. Algebraic method of nonparametric identification of abnormal modes of the power system as a dynamic MIMO system. System. J. Comput. Syst. Sci. Int., 2020, vol. 59, no. 6, pp. 845--853. DOI: https://doi.org/10.1134/S1064230720060039

[16] Gadzhiev M.G., Misrikhanov M.Sh., Ryabchenko V.N., et al. Matrichnye metody analiza i upravleniya perekhodnymi protsessami v elektroenergeticheskikh sistemakh [Matrix methods of analysis and control of transient processes in electric power systems]. Moscow, MPEI Publ., 2019.