# Invertion of infinite Gaussian matrices

• Fedorov Foma M., foma_46@mail.ru Северо-Восточный федеральный университет им. М. К. Аммосова, Научно-исследовательский институт математики, ул. Кулаковского 48, Якутск 677891
• Pavlov Nikifor N., pnn10@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677891, Russia
• Potapova Sargylana V., sargyp@inbox.ru M. K. Ammosov North-Eastern Federal University, Scientiﬁc Research Institute of Mathematics, 48 Kulakovsky Street, Yakutsk 677891, Russia
• Ivanova Oksana F., o_buskarova@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677891, Russia
Keywords: infinite system, linear algebraic equation, infinite triangular matrix, Gaussian matrix, inverse matrix, infinite determinant

### Abstract

We study existence of the left inverse, right inverse and inverse of Gaussian infinite matrices (those are the upper infinite triangular matrices with nonzero elements on the main diagonal). The existence of a unique inverse of the Gaussian matrix is proved. Also, an explicit expression for the inverse of the Gaussian matrix of any order is found, including the infinite case.
Implementation of this expression is very convenient, since calculations are based on recurrence relations. Such approach can be extended to triangular infinite matrices (those are the lower infinite triangular matrices with nonzero elements on the main diagonal). Thus, there is the possibility of inversion of an infinite matrix of infinite rank, since such matrices decompose into the product of two matrices, a triangular and a Gaussian.

### References

[1]
Kantorovich L. V. and Krylov V. I., Approximate Methods of Higher Analysis, P. Nordhoff, Groningen (1958).

[2]
Cooke R. G., Infinite Matrices and Sequence Spaces, Macmillan& Co, Ltd, London (1950).

[3]
Dienes P., “Notes on linear equations in infinite matrices,” Q. J. Math. (Oxf. Ser.), 3, 253–268 (1932).

[4]
Kagan V. F., Fundamentals of the Theory of Determinants [in Russian], Ukr. Gos. Izdat., Odessa (1922).

[5]
Gantmakher F. R., The Theory of Matrices, Nauka, Moscow (1967).

[6]
Koch H., “On regular and irregular solutions of some infinite systems of linear equations,” C. R. Scand. Congr. Math. (Stockholm, 1909), Teubner, Leipzig (1910).

[7]
Finta B., “The Gauss method for systems of linear equations (II),” Petru Maior Univ. Tg. Mures (2006) (http://www.upm.ro/InterIng2007/Papers/Sеction6/ 6-Gauss_Method_Infinite_System_2_pVI-6-1_5.pdf).

[8]
Fedorov F. M., “On Gauss algorithm for infinite systems of linear algebraic equations [in Russian],” Mat. Zamet. YAGU, 19, No. 1, 133–140 (2012).

[9]
Fedorov F. M., “Inhomogeneous Gaussian infinite systems of linear algebraic equations [in Russian],” Mat. Zamet. YAGU, 19, No. 1, 124–131 (2012).

[10]
Fedorov F. M., Pavlov N. N., and Ivanova O. F., “Algorithms implementing the decisions of infinite systems of linear algebraic equations [in Russian],” Mat. Zamet. YAGU, 20, No. 1, 215–223 (2013).

[11]
Fedorov F. M., Ivanova O. F., and Pavlov N. N., “Convergence of the method of reduction and consistency of infinite systems,” Vestn. Severo-Vostochn. Univ., 11, No. 2, 14–21 (2014).

[12]
Ivanova O. F., Pavlov N. N., and Fedorov F. M., “On the principal and strictly particular solutions to infinite systems,” Comput. Math. Math. Phys., 56, No. 3, 343–353 (2016).

[13]
Fedorov F. M., Ivanova O. F., and Pavlov N. N., “On the specificities of infinite systems,” Mat. Zamet. SVFU, 22, No. 4, 62–78 (2015).

[14]
Fedorov F. M., “On remarkable relations and the passage to the limit in the theory of infinite systems,” J. Generalized Lie Theory Appl., 9, 224 (2015).

[15]
Fedorov F. M., “On the theory of infinite systems of linear algebraic equation,” TWMS J. Pure Appl. Math., 6, No. 2, 202–212 (2015).

[16]
Fedorov F. M., Periodic Infinite Systems of Linear Algebraic Equations, Nauka, Novosibirsk (2009).

[17]
Fedorov F. M., Infinite systems of linear algebraic equations and their applications, Nauka, Novosibirsk (2011).
How to Cite
Fedorov, F., Pavlov, N., Potapova, S. and Ivanova, O. (&nbsp;) “Invertion of infinite Gaussian matrices”, Mathematical notes of NEFU, 25(3), pp. 54-67. doi: https://doi.org/10.25587/SVFU.2018.99.16951.
Issue
Section
Mathematics