The general $\phi$-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations

Abstract

Abstract. Let H m×n be the space of m × n matrices over H, where H is the real quaternion algebra. Let A φ be the n × m matrix obtained by applying φ entrywise to the transposed matrix A T , where A ∈ H m×n and φ is a nonstandard involution of H. In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix A φ are given. Two systems of mixed pairs of quaternion matrix Sylvester equationswhere Z is φ-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution (X, Y, Z) to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices are presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. Some numerical examples are provided to illustrate the main results.

Some quaternion matrix equations involving Φ-Hermicity

“…In this paper, we consider only the nonstandard involution. Some examples of nonstandard involutions can be found in [7].…”

“…Solving the real quaternion matrix equations involving φ-Hermicity is a new topic in quaternion linear algebra and has attracted more and more attention in recent years. For example, He, Liu and Tam [7] considered mixed pairs of quaternion matrix Sylvester equations involving φ-Hermicity. Very recently, He [6] considered the following system of quaternion matrix equations involving φ-Hermicity…”

Solutions to the linear transpose matrix equations and their application in control

“…In the present work, we are not concerned the numerical solutions to the linear transpose matrix equations and we suggested the readers to see (Zhou et al 2011;Wang et al 2007;Xie et al 2010) and their references therein. Besides the topics above on linear matrix equation, some other topics were also focused on: for instance, the solutions to the (generalized) Sylvester matrix equation over real field, complex field and quaternion field; for details, please see (Jiang and Wei 2003;He et al 2018;He 2014, 2013;Duan 2005, 2006;Wu et al 2009;He et al 2020He et al , 2017Wang et al 2011;Wang and Li 2009;Ma 1966). Here we should be pointed out that one of the important remarkable results is the solutions to the Sylvester matrix equation over complex or quaternion field Duan 2005, 2006;He et al 2020He et al , 2017Wang et al 2011;Wang and Li 2009;Wang and Zhang 2008).…”

Cramer’s rules for the system of quaternion matrix equations withη-Hermicity

“…An iterative algorithm for determining g (-skew)-Hermitian least-squares solutions to the quaternion matrix equation 3was established in [27]. For more related papers on g-Hermicity and its generalization, /-Hermicity, one may refer to [28][29][30][31][32][33][34][35][36][37][38].…”