Abstract
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 = ±1. By an eignvalue decomposition of P, the constrained problem can be equivalently transformed to a well-known unconstrained problem of 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.
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References
Baksalary J K, Kala R. The matrix equation AXB + CYD = E, Linear Algebra Appl, 1980, 30: 141–147.
Chu D, Moor B D. On a varitional formulation of the QSVD and RSVD, Linear Algebra Appl, 2000, 311: 61–78.
Chu K E. Singular value and generalized singular value decompositions and solutions of linear matix equations, Linear Algebra Appl, 1987, 88/89: 83–98.
Chu K E. Symmetric solutions of linear matrix equations by matrix decompositions, Linear Algebra Appl, 1989, 119: 35–50.
Dai H. On the symmetric solutions of linear matrix equations, Linear Algebra Appl, 1990, 131: 1–7.
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.
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.
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.
Özgüler A B. The equation AXB + CYD = E over a principle ideal domain, SIAM J Matrix Anal Appl, 1991, 12: 581–591.
Paige C C, Saunders M A. Towards a generalized singular value decomposition, SIAM J Numer Anal, 1981, 18: 398–405.
Paige C C. Computing the generalized singular value decomposition, SIAM J Sci Comput, 1986, 7: 1126–1146.
Xu G P, Wei M S, Zheng D S. On solutions of matrix equation AXB + CYD = F, Linear Algebra Appl, 1998, 279: 93–109.
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The work of the first author was supported by the Young Talent Foundation of Zhejiang Gongshang University.
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Qiu, Y., Qiu, C. Matrix equation AXB = E with PX = sXP constraint. Appl. Math.- J. Chin. Univ. 22, 441–448 (2007). https://doi.org/10.1007/s11766-007-0409-9
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DOI: https://doi.org/10.1007/s11766-007-0409-9