A simultaneous decomposition of seven matrices over real quaternion algebra and its applications

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Let H be the real quaternion algebra and Hⁿ× ᵐ denote the set of all n× m matrices over H. In this paper, we construct a simultaneous decomposition of seven general real quaternion matrices with compatible sizes: A∈ Hᵐ× ⁿ, B∈ Hᵐ× ᵖ₁,C∈ Hᵐ× ᵖ₂,D∈ Hᵐ× ᵖ₃,E∈ Hq₁× ⁿ,F∈ Hq₂× ⁿ,G∈ Hq₃× ⁿ. As applications of the simultaneous matrix decomposition, we give solvability conditions, general solutions, as well as the range of ranks of the general solutions to the following two real quaternion matrix equations BXE+CYF+DZG=A and BX+WE+CYF+DZG=A, where A,B,C,D,E,F, and G are given real quaternion matrices.