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.

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How to Cite
Fedorov, F., Pavlov, N., Potapova, S. and Ivanova, O. ( ) “Invertion of infinite Gaussian matrices”, Mathematical notes of NEFU, 25(3), pp. 54-67. doi: https://doi.org/10.25587/SVFU.2018.99.16951.
Section
Mathematics