Simultaneous solutions of matrix equations and simultaneous equivalence of matrices
Simultaneous solutions of matrix equations and simultaneous equivalence of matrices
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DOI：
10.1016/j.laa.2012.06.004
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发表时间：
2012-11
影响因子：
1.1
通讯作者：
Sang-Gu Lee;Quoc-Phong Vu
中科院分区：
文献类型：
--
作者：
Sang-Gu Lee;Quoc-Phong Vu
给定域F上相应维数的矩阵Ai，Bi和Ci（i∈I），证明了：（i）若AiCiOBi同时相似于AiOOBi，则矩阵西尔维斯特方程AiX-XBi=Ci存在联立解X;（ii）若AiCiOBi同时等价于AiOOBi，则矩阵方程AiX-YBi=Ci存在联立解X，Y.我们还证明了对于混合矩阵西尔维斯特方程A1 X1-YB 1 =C1，A2 X2-YB 2 = C2和广义Stein方程X-AYB=C，类似的结果也成立.
Given matrices Ai, Biand Ci(i∈I) of corresponding dimensions over a field F, we prove that: (i) if AiCiOBiare simultaneously similar to AiOOBi, then there exists a simultaneous solution X to the matrix Sylvester equations AiX-XBi=Ci; and (ii) if AiCiOBiare simultaneously equivalent to AiOOBi, then there exist simultaneous solutions X,Y to the matrix equations AiX-YBi=Ci. We also show that analogous results hold for mixed pairs of matrix Sylvester equations A1X1-YB1=C1, A2X2-YB2=C2and for generalized Stein equations X-AYB=C.