Recent zbMATH articles in MSC 15A09 https://zbmath.org/atom/cc/15A09 2022-06-24T15:10:38.853281Z Werkzeug Algebraic conditions for the solvability of system of three linear equations in a ring https://zbmath.org/1485.15007 2022-06-24T15:10:38.853281Z "Milošević, Jovana" https://zbmath.org/authors/?q=ai:milosevic.jovana Summary: In this paper, we give algebraic conditions for the existence of a solution and the expression for the general solution of the system of the equations $a_1xb_1=c_1, \quad a_2xb_2=c_2, \quad a_3xb_3=c_3,$ when each element belongs to a ring with a unit. As an application, we get necessary and sufficient conditions for the existence of common inner inverse of three regular elements. Weighted inner inverse for rectangular matrices https://zbmath.org/1485.15008 2022-06-24T15:10:38.853281Z "Behera, Ratikanta" https://zbmath.org/authors/?q=ai:behera.ratikanta "Mosić, Dijana" https://zbmath.org/authors/?q=ai:mosic.dijana "Sahoo, Jajati Kesahri" https://zbmath.org/authors/?q=ai:sahoo.jajati-kesahri "Stanimirović, Predrag S." https://zbmath.org/authors/?q=ai:stanimirovic.predrag-s Summary: To extend the notation of inner inverses, we define weighted inner inverses of a rectangular matrix. In particular, we introduce a $$W$$-weighted $$(B, C)$$-inner inverse of $$A$$, for given matrices \textit{A, W, B, C}, and present some characterizations and conditions for its existence. Since this new inverse is not unique, we describe the set of all $$W$$-weighted $$(B, C)$$-inner inverses of a given matrix. Several invertible matrix expressions which involve a $$W$$-weighted $$(B, C)$$-inner inverse of $$A$$ are studied. Using such expressions, we represent a $$W$$-weighted $$(B, C)$$-inner inverse of some other matrix $$E$$. As a particular case of $$W$$-weighted $$(B, C)$$-inner inverse, we investigate $$(B, C)$$-inner inverses of a rectangular matrix. We also establish some reverse order law properties considering weighted inner inverses. Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices https://zbmath.org/1485.15009 2022-06-24T15:10:38.853281Z "Wei, Yunlan" https://zbmath.org/authors/?q=ai:wei.yunlan "Jiang, Xiaoyu" https://zbmath.org/authors/?q=ai:jiang.xiaoyu "Jiang, Zhaolin" https://zbmath.org/authors/?q=ai:jiang.zhaolin "Shon, Sugoog" https://zbmath.org/authors/?q=ai:shon.sugoog Summary: In this paper, we deal mainly with a class of periodic tridiagonal Toeplitz matrices with perturbed corners. By matrix decomposition with the Sherman-Morrison-Woodbury formula and constructing the corresponding displacement of matrices we derive the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type $$I$$ in the form of products of Fermat numbers and some initial values. Furthermore, the properties of type $$II$$ matrix can be also obtained, which benefits from the relation between type $$I$$ and $$II$$ matrices. Finally, we propose two algorithms for computing these properties and make some analysis about them to illustrate our theoretical results. Extension of Moore-Penrose inverse of tensor via Einstein product https://zbmath.org/1485.15035 2022-06-24T15:10:38.853281Z "Panigrahy, Krushnachandra" https://zbmath.org/authors/?q=ai:panigrahy.krushnachandra "Mishra, Debasisha" https://zbmath.org/authors/?q=ai:mishra.debasisha Summary: The notion of the Moore-Penrose inverse of an even-order tensor and the two-term reverse-order law for the Moore-Penrose inverse of even-order tensors via the Einstein product were introduced, very recently. In this article, the Moore-Penrose inverse of an arbitrary tensor is introduced first. Various new expressions of the Moore-Penrose inverse are also proposed. A new generalized inverse of a tensor called \textit{product Moore-Penrose inverse} is then introduced by extending the Moore-Penrose inverse of an arbitrary tensor. A necessary and sufficient condition for the coincidence of the Moore-Penrose inverse and the product Moore-Penrose inverse of an arbitrary tensor is also provided. Prior to these, a set of new sufficient conditions for computing the Moore-Penrose inverse of the product of two tensors via the Einstein product is illustrated. Finally, some necessary and sufficient conditions are obtained for the three-term reverse-order law of tensors via the same product. A fixed point method to solve linear operator equations involving self-adjoint operators in Hilbert space https://zbmath.org/1485.47015 2022-06-24T15:10:38.853281Z "Khan, Mohammad Saeed" https://zbmath.org/authors/?q=ai:khan.mohammad-saeed "Teodorescu, Dinu" https://zbmath.org/authors/?q=ai:teodorescu.dinu Summary: In this paper, we provide existence and uniqueness results for linear operator equations of the form $$(I+A^m)x=f$$, where $$A$$ is a self-adjoint operator in Hilbert space. Some applications to the study of invertible matrices are also presented. A new approach to solve linear systems https://zbmath.org/1485.65027 2022-06-24T15:10:38.853281Z "Kameli, A." https://zbmath.org/authors/?q=ai:kameli.a "Jafari, H." https://zbmath.org/authors/?q=ai:jafari.hossein "Moradi, A." https://zbmath.org/authors/?q=ai:moradi.abbas|moradi.azam|moradi.afsaneh|moradi.ahmad|moradi.a-r|moradi.amir|moradi.abed|moradi.afshin|moradi.ali|moradi.akbar Summary: Solving linear system is central to scientific computations. Given a linear system $$Ax=b$$, where $$A$$ is a non-singular real matrix. None of the existing solution approaches to the system is capable of exploiting special information provided by a proper sub-system. Only recently, \textit{A. Moradi} et al. [Nonlinear Dyn. Syst. Theory 19, No. 1, 193--199 (2019; Zbl 07104581)] showed that in the special case where corner minors of $$A$$ are all non-zero, solutions to sub-systems generated by the corner sub-matrices of $$A$$ could be used to construct a solution to the original system. In this paper, we extend the result to the case where no restriction is imposed on the coefficient matrix, $$A$$. A fast computational algorithm for computing outer pseudo-inverses with numerical experiments https://zbmath.org/1485.65043 2022-06-24T15:10:38.853281Z "Dehghan, Mehdi" https://zbmath.org/authors/?q=ai:dehghan.mehdi "Shirilord, Akbar" https://zbmath.org/authors/?q=ai:shirilord.akbar Summary: In many studies in applied sciences and engineering one should find outer pseudo-inverse of a matrix. In this paper, we propose a new efficient algorithm for computing the outer pseudo-inverse of a matrix. We study the convergence analysis of the new algorithm. Finally, test problems and simulation results support the theoretical approach. The least-squares solutions in linear codes based multisecret-sharing approach https://zbmath.org/1485.94153 2022-06-24T15:10:38.853281Z "Çalkavur, Selda" https://zbmath.org/authors/?q=ai:calkavur.selda "Nauman, Syed Khalid" https://zbmath.org/authors/?q=ai:nauman.syed-khalid "Özel, Cenap" https://zbmath.org/authors/?q=ai:ozel.cenap "Zekraoui, Hanifa" https://zbmath.org/authors/?q=ai:zekraoui.hanifa Summary: The theory of generalised inverses of matrices has been applied in many areas since 1950. Some of them are Markov chains, robotics differential equations, etc. Especially, the applications over finite fields of this theory have an important role in cryptography and coding theory. In this work, we study a new multisecret-sharing scheme by using the generalised inverses of matrices and least-squares solutions in linear codes. We determine the access structure. Its security improves on that of multisecret-sharing schemes.