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Matrix equation \(AXB=E\) with \(PX=sXP\) constraint. (English) Zbl 1150.15008
Summary: The matrix equation \(AXB=E\) with the constraint \(PX=sXP\) is considered, where \(P\) is a given Hermitian matrix satisfying \(P^2=I\) and \(s=\pm 1\). By an eigenvalue decomposition of \(P\), the constrained problem can be equivalently transformed to a well-known unconstrained problem of a matrix equation whose coefficient matrices contain the corresponding eigenvector, and hence the constrained problem can be solved in terms of the eigenvectors of \(P\). A simple and eigenvector-free formula of the general solutions to the constrained problem by generalized inverses of the coefficient matrices \(A\) and \(B\) is presented. Moreover, a similar problem of the matrix equation with generalized constraint is discussed.

15A24 Matrix equations and identities
15A18 Eigenvalues, singular values, and eigenvectors
15A09 Theory of matrix inversion and generalized inverses
Full Text: DOI
[1] Baksalary J K, Kala R. The matrix equation AXB + CYD = E, Linear Algebra Appl, 1980, 30: 141–147. · Zbl 0437.15005
[2] Chu D, Moor B D. On a varitional formulation of the QSVD and RSVD, Linear Algebra Appl, 2000, 311: 61–78. · Zbl 0970.65037
[3] Chu K E. Singular value and generalized singular value decompositions and solutions of linear matix equations, Linear Algebra Appl, 1987, 88/89: 83–98. · Zbl 0612.15003
[4] Chu K E. Symmetric solutions of linear matrix equations by matrix decompositions, Linear Algebra Appl, 1989, 119: 35–50. · Zbl 0688.15003
[5] Dai H. On the symmetric solutions of linear matrix equations, Linear Algebra Appl, 1990, 131: 1–7. · Zbl 0712.15009
[6] Deng Y B, Hu X Y, Zhang L. Least squares solutions of BXA T = T over symmetric, skewsymmetric, and positive semidefinite X*, SIAM J Matrix Anal Appl, 2003, 25: 486–494. · Zbl 1050.65037
[7] Liao A P, Bai Z Z, Lei Y. Best approximate solution of matrix equation AXB + CY D = E, SIAM J Matrix Anal Appl, 2005, 27: 675–688. · Zbl 1096.15004
[8] Moor B D, Golub G H. Generalized singular value decompositions:a proposal for a standardized nomenclature, Zaterual Report, 89-10, ESAT-SISTA, Leuven, Belgium, 1989.
[9] Özgüler A B. The equation AXB + CYD = E over a principle ideal domain, SIAM J Matrix Anal Appl, 1991, 12: 581–591. · Zbl 0742.15006
[10] Paige C C, Saunders M A. Towards a generalized singular value decomposition, SIAM J Numer Anal, 1981, 18: 398–405. · Zbl 0471.65018
[11] Paige C C. Computing the generalized singular value decomposition, SIAM J Sci Comput, 1986, 7: 1126–1146. · Zbl 0621.65030
[12] Xu G P, Wei M S, Zheng D S. On solutions of matrix equation AXB + CYD = F, Linear Algebra Appl, 1998, 279: 93–109. · Zbl 0933.15024
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