Explicit Solutions of the Singular Yang–Baxter-like Matrix Equation and Their Numerical Computation

Kumar, Ashim; Cardoso, João R.; Singh, Gurjinder

doi:10.1007/s00009-022-01982-y

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…Implementing those computational procedures in symbolic form, in the form of exact arithmetic and doubleprecision arithmetic is also a concern. Thus, this paper has greatly extended the previous work of [33].…”

Section: Discussionsupporting

confidence: 52%

“…Moreover, corresponding to each B satisfying (5), the general solution to ( 6) is equal to (4). Furthermore, in ( [33], [Lemma 3.1]), it was shown that X 0 solves (1) iff there exists an appropriate matrix B which ensures that X 0 is a solution to (6). This proves the lemma.…”

mentioning

confidence: 76%

“…The first statement is verified. For the given choice of B, it follows from [33] (Lemma 4.1) that the matrix B satisfies the conditions in (5). So by Lemma 3 (resp.…”

Section: Lemma 3 Assumementioning

confidence: 92%

“…This paper contributes to resolving these issues. Recall that in [33], the authors have developed effective formulae for calculating infinitely many solutions using finite-precision arithmetic. Nevertheless, those methods do not perform well when A is an ill-conditioned or a matrix with multivariate rational functional entries.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

et al. 2022

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This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as {2,5}-inverses) of a matrix M, which commute with M. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as {2,5}-inverses) of a matrix, which commute with it. This class of {2,5}-inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments.

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Section: Discussionsupporting

confidence: 52%

mentioning

confidence: 76%

“…The first statement is verified. For the given choice of B, it follows from [33] (Lemma 4.1) that the matrix B satisfies the conditions in (5). So by Lemma 3 (resp.…”

Section: Lemma 3 Assumementioning

confidence: 92%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

et al. 2022

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Zeroing Neural Network Approaches Based on Direct and Indirect Methods for Solving the Yang–Baxter-like Matrix Equation

et al. 2022

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This research introduces three novel zeroing neural network (ZNN) models for addressing the time-varying Yang–Baxter-like matrix equation (TV-YBLME) with arbitrary (regular or singular) real time-varying (TV) input matrices in continuous time. One ZNN dynamic utilizes error matrices directly arising from the equation involved in the TV-YBLME. Moreover, two ZNN models are proposed using basic properties of the YBLME, such as the splitting of the YBLME and sufficient conditions for a matrix to solve the YBLME. The Tikhonov regularization principle enables addressing the TV-YBLME with an arbitrary input real TV matrix. Numerical experiments, including nonsingular and singular TV input matrices, show that the suggested models deal effectively with the TV-YBLME.

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