Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices

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• 作者：Zhuo-Heng He^a ; ^b ; ^{hzh19871126@126.com" class="auth_mail" title="E-mail the corresponding author} ; Oscar Mauricio Agudelo^b ; ^{mauricio.agudelo@esat.kuleuven.be" class="auth_mail" title="E-mail the corresponding author} ; Qing-Wen Wang^a ; ^{wqw@t.shu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{wqw369@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; Bart De Moor^b ; ^{bart.demoor@esat.kuleuven.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A03 ; 15A21 ; 15A23 ; 15A24
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：496
• 期：Complete
• 页码：549-593
• 全文大小：635 K

文摘

In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices: A_i∈C^p_i×t_i$A_i\in {\mathbb{C}}^{p_i\times t_i}$, 34b1b13f9e959b0b721eaa4c863" title="Click to view the MathML source">B_i∈C^s_i×q_i$B_i\in {\mathbb{C}}^{s_i\times q_i}$, C_i∈C^p_i×t_i+1$C_i\in {\mathbb{C}}^{p_i\times t_{i+1}}$, D_i∈C^s_i+1×q_i$D_i\in {\mathbb{C}}^{s_{i+1}\times q_i}$, and E_i∈C^p_i×q_i$E_i\in {\mathbb{C}}^{p_i\times q_i}$ (b40e2a2ce0" title="Click to view the MathML source">i=1,2,3$i=1,2,3$). We show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two-sided coupled generalized Sylvester matrix equations with four unknowns A_iX_iB_i+C_iX_i+1D_i=E_i$A_i{X}_i{B}_i+C_i{X}_{i+1}{D}_i=E_i$ (b40e2a2ce0" title="Click to view the MathML source">i=1,2,3$i=1,2,3$). Apart from proving an expression for the general solutions to this system, we derive the range of ranks of these solutions using the ranks of the given matrices A_i$A_i$, B_i$B_i$, C_i$C_i$, D_i$D_i$, and E_i$E_i$. We provide some numerical examples to illustrate our results. Moreover, we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k   two-sided coupled generalized Sylvester matrix equations with k+1$k+1$ unknowns A_iX_iB_i+C_iX_i+1D_i=E_i$A_i{X}_i{B}_i+C_i{X}_{i+1}{D}_i=E_i$ (i=1,…,k$i=1,\dots ,k$, k≥4$k\ge 4$). The main results are also valid over the real number field and the real quaternion algebra.