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: class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si1.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=75b775080be489fccb121798ba85f9e6" title="Click to view the MathML source">A_i∈C^p_i×t_iclass="mathContainer hidden">class="mathCode">$A_i\in {\mathbb{C}}^{p_i\times t_i}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si173.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=8664f34b1b13f9e959b0b721eaa4c863" title="Click to view the MathML source">B_i∈C^s_i×q_iclass="mathContainer hidden">class="mathCode">$B_i\in {\mathbb{C}}^{s_i\times q_i}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si174.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=e322ebd3ea6c7fc83782f4093296ea4f" title="Click to view the MathML source">C_i∈C^p_i×t_i+1class="mathContainer hidden">class="mathCode">$C_i\in {\mathbb{C}}^{p_i\times t_{i+1}}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si175.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=05a83a7edf47ba9c4353a3b4105ab270" title="Click to view the MathML source">D_i∈C^s_i+1×q_iclass="mathContainer hidden">class="mathCode">$D_i\in {\mathbb{C}}^{s_{i+1}\times q_i}$, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si118.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=a1e555735438e628a8639e766be1f4d0" title="Click to view the MathML source">E_i∈C^p_i×q_iclass="mathContainer hidden">class="mathCode">$E_i\in {\mathbb{C}}^{p_i\times q_i}$ (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si6.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=d9073246ddab5373b26400b40e2a2ce0" title="Click to view the MathML source">i=1,2,3class="mathContainer hidden">class="mathCode">$i=1,2,3$). 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si7.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=ca7629c7d319581155e054a1c052c270" title="Click to view the MathML source">A_iX_iB_i+C_iX_i+1D_i=E_iclass="mathContainer hidden">class="mathCode">$A_i{X}_i{B}_i+C_i{X}_{i+1}{D}_i=E_i$ (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si6.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=d9073246ddab5373b26400b40e2a2ce0" title="Click to view the MathML source">i=1,2,3class="mathContainer hidden">class="mathCode">$i=1,2,3$). 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si8.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=622f0fa1b0e5d254498789501ca8b628" title="Click to view the MathML source">A_iclass="mathContainer hidden">class="mathCode">$A_i$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si9.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=d8c50601376c01616a315d2ffdc8a17b" title="Click to view the MathML source">B_iclass="mathContainer hidden">class="mathCode">$B_i$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si10.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=b7c6e4b7802281a42ea68fed2b6fc2c5" title="Click to view the MathML source">C_iclass="mathContainer hidden">class="mathCode">$C_i$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si11.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=8d667ba8cf6f3bfe48cdd561653cd2ea" title="Click to view the MathML source">D_iclass="mathContainer hidden">class="mathCode">$D_i$, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si12.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=c4591b24e1fc5d8328ae9526965c7dbe" title="Click to view the MathML source">E_iclass="mathContainer hidden">class="mathCode">$E_i$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si13.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=e7f3889d091522aa3c73d49cdbe33215" title="Click to view the MathML source">k+1class="mathContainer hidden">class="mathCode">$k+1$ unknowns class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si7.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=ca7629c7d319581155e054a1c052c270" title="Click to view the MathML source">A_iX_iB_i+C_iX_i+1D_i=E_iclass="mathContainer hidden">class="mathCode">$A_i{X}_i{B}_i+C_i{X}_{i+1}{D}_i=E_i$ (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si14.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=b1332cfc5ea8f449c3b598cf84082f27" title="Click to view the MathML source">i=1,…,kclass="mathContainer hidden">class="mathCode">$i=1,\dots ,k$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516001142&_mathId=si15.gif&_user=111111111&_pii=S0024379516001142&_rdoc=1&_issn=00243795&md5=f89115cba1a45f55a15848a3924a8774" title="Click to view the MathML source">k≥4class="mathContainer hidden">class="mathCode">$k\ge 4$). The main results are also valid over the real number field and the real quaternion algebra.