A system of generalized Sylvester quaternion matrix equations and its applications

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• 作者：Xiang Zhang ; ^{zxjnsc@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：System of matrix equation ; Quaternion ; General solution ; η-Hermitian ; Moore&ndash ; Penrose inverse
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：273
• 期：Complete
• 页码：74-81
• 全文大小：308 K

文摘

Let Hm×n${\mathbb{H}}^{m\times n}$ be the set of all m × n   matrices over the real quaternion algebra. We call that A∈Hn×n$A\in {\mathbb{H}}^{n\times n}$ is η  -Hermitian if A=−ηA^*η,$A=-\mathrm{<font\; color="red">\eta </font>}{A}^{*}\mathrm{<font\; color="red">\eta </font>},$η ∈ {i, j, k}, where i, j, k   are the quaternion units. Denote Aη*=−ηA^*η$A^{\mathrm{<font\; color="red">\eta </font>}*}=-\mathrm{<font\; color="red">\eta </font>}{A}^{*}\mathrm{<font\; color="red">\eta </font>}$. In this paper, we derive some necessary and sufficient conditions for the solvability to the system of generalized Sylvester real quaternion matrix equations A_iX_i+Y_iB_i+C_iZD_i=E_i,(i=1,2),$A_i{X}_i+Y_i{B}_i+C_{i}Z{D}_i=E_i,(i=1,2),$ and give an expression of the general solution to the above mentioned system. As applications, we give some solvability conditions and general solution for the generalized Sylvester real quaternion matrix equation A₁X+YB₁+C₁ZD₁=E₁,$A_{1}X+Y{B}_1+C_{1}Z{D}_1=E_1,$ where Z is required to be η-Hermitian. We also present some solvability conditions and general solution for the system of real quaternion matrix equations involving η  -Hermicity $A_i{X}_i+{\left(A_i{X}_i\right)}^{\mathrm{<font\; color="red">\eta </font>}*}+B_{i}Y{B}_i^{\mathrm{<font\; color="red">\eta </font>}*}=C_i,(i=1,2),$ where Y is required to be η-Hermitian. Our results include some well-known results as special cases.