Sylvester-type quaternion matrix equations with arbitrary equations and arbitrary unknowns

05/30/2020

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by   Zhuo-Heng He, et al.

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In this paper, we prove a conjecture which was presented in a recent paper [Linear Algebra Appl. 2016; 496: 549–593]. We derive some practical necessary and sufficient conditions for the existence of a solution to a system of coupled two-sided Sylvester-type quaternion matrix equations with arbitrary equations and arbitrary unknowns A_iX_iB_i+C_iX_i+1D_i=E_i, i=1,k. As an application, we give some practical necessary and sufficient conditions for the existence of an η-Hermitian solution to the system of quaternion matrix equations A_iX_iA^η*_i+C_iX_i+1C^η*_i=E_i in terms of ranks, i=1,k.