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The reflexive and anti-reflexive solutions of the matrix equation $$AX=B$$. (English) Zbl 1050.15016
Let $$P\in\mathbb{C}^{n\times n}$$ be both Hermitian and unitary. A matrix $$X\in\mathbb{C}^{n\times n}$$ is said to be reflexive (resp. anti-reflexive) with respect to $$P$$ when $$X= PXP$$ (resp. $$X = -PXP$$). Conditions are given for existence of reflexive and antireflexive solutions (with respect to $$P$$) of the equation (1) $$AX= B$$, where $$A,B\in\mathbb{C}^{m\times n}$$ are given matrices. If $$X_0\in \mathbb{C}^{n\times n}$$ is a given matrix, expressions are given for the reflexive and anti-reflexive solutions of (1) which are the nearest to $$X_0$$ in the Frobenius norm.

##### MSC:
 15A24 Matrix equations and identities
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##### References:
  Andersson, L.E.; Elfving, T., A constrained procrustes problem, SIAM J. matrix anal. appl., 18, 124-139, (1997) · Zbl 0880.65017  Chen, J.L.; Chen, X.H., Special matrices, (2001), Qinghua University Press Beijing, China, (in Chinese)  Chen, H.C., Generalized reflexive matrices: special properties and applications, SIAM J. matrix anal. appl., 19, 140-153, (1998) · Zbl 0910.15005  H.C. Chen, The SAS Domain Decomposition Method for Structural Analysis, CSRD Tech. report 754, Center for Supercomputing Research and Development, University of Illinois, Urbana, IL, 1988  Chen, H.C.; Sameh, A., Numerical linear algebra algorithms on the ceder system, (), 101-125  Cheney, E.W., Introduction to approximation theory, (1966), McGraw-Hill New York · Zbl 0161.25202  Chu, K-W.E., Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear algebra appl., 88, 89-98, (1987) · Zbl 0612.15003  Chu, K-W.E., Symmetric solutions of linear matrix equations by matrix decompositions, Linear algebra appl., 119, 35-50, (1989) · Zbl 0688.15003  Dai, H., On the symmetric solutions of linear matrix equations, Linear algebra appl., 131, 1-7, (1990) · Zbl 0712.15009  Don, F.J.H., On the symmetric solutions of a linear matrix equation, Linear algebra appl., 93, 1-7, (1987) · Zbl 0622.15001  Higham, N.J., The symmetric procrustes problem, Bit, 28, 133-143, (1988) · Zbl 0641.65034  Mgnus, J.R.; Neudecker, H., The commutation matrix: some properties and applications, Ann. statist., 7, 381-394, (1979) · Zbl 0414.62040  Mgnus, J.R.; Neudecker, H., The elimination matrix: some lemmas and applications, SIAM J. algebr. discrete methods, 1, 422-449, (1980) · Zbl 0497.15014  Vetter, W.J., Vector structures and solutions of linear matrix equations, Linear algebra appl., 10, 181-188, (1975) · Zbl 0307.15003  Wu, L., The re-positive defininte solutions to the matrix inverse problem AX=B, Linear algebra appl., 174, 145-151, (1992) · Zbl 0754.15003  Xie, D.X.; Zhang, L.; Hu, X.Y., The inverse problem for bisymmetric matrices on a linear manifold, Math. numer. sinica, 2, 129-138, (2000)  Zhou, S.Q.; Dai, H., The algebraic inverse eigenvalue problem, (1991), Henan Science and Technology Press Zhengzhou, China, (in Chinese)
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