Eigenvector-free solutions to $$AX = B$$ with $$PX = XP$$ and $$X^H = sX$$ constraints.(English)Zbl 1221.15025

The matrix equation $$AX=B$$ with $$PX=XP$$ and $$X^H=sX$$ is considered, where $$P$$ is a given Hermitian involutory matrix and $$s=+1$$.
It is shown that by an eigenvalue decomposition of $$P$$ one obtains a pair of similar problems which is equivalent with the initial one.
Necessary and sufficient conditions for the existence of a solution are provided, as well as a representation of the general solution.
Similar results are derived for eigenvalue-free solutions to the considered constrained problem.
The optimal approximation problem $$\min\|X-E\|_F$$ is studied, where $$E\in\mathbb{C}^{n\times n}$$ is a given matrix and $$X_s$$ is the solution set of the constrained problem.
Eigenvalue-involved and eigenvalue-free solutions for the optimal problem are provided.
A similar approach is presented for the extended problem of the matrix equations $$AX=B,\;XC=D,\;A,B\in\mathbb{C}^{m\times n},\;C,D\in\mathbb{C}^{n\times p}$$, with the same constraints.
Some numerical examples illustrate the effectiveness of the proposed methods.

MSC:

 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses 65F30 Other matrix algorithms (MSC2010)
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References:

 [1] Chen, H.C., Generalized reflexive matrices: special properties and applications, SIAM J. matrix anal. appl., 19, 1, 140-153, (1998) · Zbl 0910.15005 [2] Chen, H.C.; Sameh, A., A matrix decomposition method for orthotropic elasticity problems, SIAM J. matrix anal. appl., 10, 39-64, (1989) · Zbl 0669.73010 [3] Chen, K.E., Singular value and generalized singular value decomposition and the solution of linear matrix equations, Linear algebra appl., 88/89, 83-98, (1987) [4] Dai, H., On the symmetric solutions of linear matrix equations, Linear algebra appl., 131, 1-7, (1990) · Zbl 0712.15009 [5] Don, F.J.H., On the symmetric solutions of a linear matrix equations, Linear algebra appl., 93, 1-7, (1987) [6] Li, F.; Hu, X.Y.; Zhang, L., The generalized reflexive solutions for a class of matrix equations (AX=B,XC=D), Acta math. sci., 28, 1, 185-193, (2008) · Zbl 1150.15006 [7] Meng, C.J.; Hu, X.Y.; Zhang, L., The skew-symmetric orthogonal solutions of the matrix equation AX=B, Linear algebra appl., 402, 303-318, (2005) · Zbl 1128.15301 [8] Meng, C.J.; Hu, X.Y., The inverse problem of symmetric orthogonal matrices and its optimal approximation, Math. numer. sin., 28, 3, 269-280, (2006) [9] Peng, Z.Y., The inverse problem for Hermitian anti-reflexive matrices and its approximation, Appl. math. comput., 162, 1377-1389, (2005) · Zbl 1065.65057 [10] Peng, Z.Y.; Hu, X.Y., The reflexive and anti-reflexive solutions of the matrix equation AX=B, Linear algebra appl., 375, 147-155, (2003) · Zbl 1050.15016 [11] Paige, C.C., Computing the generalized singular value decomposition, SIAM J. sci. stat. comput., 7, 1126-1146, (1986) · Zbl 0621.65030 [12] Paige, C.C.; Saunders, M.A., Towards a generalized singular value decomposition, SIAM J. numer. anal., 18, 398-405, (1981) · Zbl 0471.65018 [13] Qiu, Y.; Zhang, Z.; Lu, J., The matrix equations AX=B, XC=D with PX=sxp constraint, Appl. math. comput., 189, 1428-1434, (2007) · Zbl 1124.15009 [14] Sun, J., Two kinds of inverse eigenvalue problems for real symmetric matrices, Math. numer. sinca., 3, 282-290, (1988) · Zbl 0656.65041 [15] Sun, J., Backward perturbation analysis of certain characteristic subspaces, Numer. math., 65, 357-393, (1993) · Zbl 0791.65023 [16] Sun, J., Matrix perturbation analysis, (2001), Science Press Beijing, pp. 50-52 [17] Tisseur, F., A chart of backward errors for singly and doubly structured eigenvalue problems, SIAM J. matrix anal. appl., 24, 877-897, (2003) · Zbl 1044.65030
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