Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra

Let Ω be a finite dimensional central algebra with an involutorial antiautomorphism σ and char Ω≠2, Ωn×n be the set of all n×n matrices over Ω. A=(aij)∈Ωn×n is called bisymmetric if aij=an−i+1,n−j+1=σ(aji) and biskewsymmetric if aij=−an−i+1,n−j+1=−σ(aji). The following systems of generalized Sylvester equations over Ω[λ]: A1X−YB1=C1, (I)⋮ AsX−YBs=Cs, A1XB1−C1XD1=E1, (II)⋮ AsXBs−CsXDs=Es, are considered. Necessary and sufficient conditions are given for the existence of constant solutions with bi(skew)symmetric constrains to (I) and (II). As a particular case, auxiliary results dealing with the system of Sylvester equations are also presented.

[1]  Harald K. Wimmer,et al.  Consistency of a pair of generalized Sylvester equations , 1994, IEEE Trans. Autom. Control..

[2]  W. Gustafson,et al.  On matrix equivalence and matrix equations , 1979 .

[3]  Russell M. Reid,et al.  Classroom Note: Some Eigenvalue Properties of Persymmetric Matrices , 1997, SIAM Rev..

[4]  J. Demmel.  Computing stable eigendecompositions of matrices , 1986 .

[5]  R. Guralnick.  Roth's theorems and decomposition of modules , 1981 .

[6]  J. Weaver.  Centrosymmetric (Cross-Symmetric) Matrices, Their Basic Properties, Eigenvalues, and Eigenvectors , 1985 .

[7]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[8]  Harald K. Wimmer.  The matrix equation X − AXB = C and an analogue of Roth's theorem , 1988 .

[9]  Robert M. Guralnick,et al.  Roth's theorems for sets of matrices , 1985 .

[10]  W. Gustafson,et al.  Roth's theorems over commutative rings , 1979 .

[11]  V. Hernández,et al.  Explicit solution of the matrix equation AXB − CXD = E , 1989 .

[12]  Michael A. Epton.  Methods for the solution ofAXD−BXC=E and its application in the numerical solution of implicit ordinary differential equations , 1980 .

[13]  Jianzhou Liu,et al.  The extension of Roth's theorem for matrix equations over a ring , 1997 .

[14]  Linear matrix equations: The module theoretic approach , 1989 .

[15]  R. Hartwig.  Roth’s equivalence problem in unit regular rings , 1976 .

[16]  Huang Liping.  The matrix equation AXB−GXD = E over the quaternion field , 1996 .

[17]  W. E. Roth,et al.  The equations $AX-YB=C$ and $AX-XB=C$ in matrices , 1952 .

[18]  A. Cantoni,et al.  Eigenvalues and eigenvectors of symmetric centrosymmetric matrices , 1976 .

[19]  VLADIMÍR KUČERA,et al.  Constant solutions of polynomial equations , 1991 .

[20]  Roth's removal rule revisited , 1983 .

[21]  Harald K. Wimmer.  Roth's theorems for matrix equations with symmetry constraints , 1994 .

[22]  R. Guralnick.  Matrix equivalence and isomorphism of modules , 1982 .

[23]  Alan L. Andrew.  Classroom Note: Centrosymmetric Matrices , 1998, SIAM Rev..