## An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation $$AXB=C$$.(English)Zbl 1146.65036

The authors introduce a simple algorithm to decide whether the matrix equation $$AXB=C$$ with compatibly dimensioned given matrices $$A, B$$, and $$C$$ has a skew-symmetric solution $$X$$. The algorithm finds this solution – if possible – in finitely many iterations for any given error bound. Moreover the same matrix equation for a modified right hand side $$\tilde C$$ and $$\tilde X$$ can be used to find the minimal norm skew-symmetric solution $$X$$ to the original $$AXB=C$$ equation. Both algorithms work for any skew-symmetric starting matrix.

### MSC:

 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities

### Keywords:

matrix equation; skew-symmetric solution; iterative method
Full Text:

### References:

  Bjerhammer, A., Rectangular reciprocal matrices with special reference to geodetic calculations, Kung. tekn. hogsk. handl. Stockholm, 45, 1-86, (1951)  Eric Chu, K.W., Symmetric solutions of linear matrix equations by matrix decompositions, Linear algebra appl., 119, 35-50, (1989) · Zbl 0688.15003  Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), John Hopkins University Press Baltimore, MD · Zbl 0865.65009  Henk Don, F.J., On the symmetric solution of a linear matrix equation, Linear algebra appl., 93, 1-7, (1988) · Zbl 0622.15001  Hua, D., On the symmetric solutions of linear matrix equations, Linear algebra appl., 131, 1-7, (1990) · Zbl 0712.15009  Magnus, J.R., L-structured matrices and linear matrix equation, Linear multilinear algebra appl., 14, 67-88, (1983) · Zbl 0527.15006  Mitra, S.K., Common solutions to a pair of linear matrix equations $$A_1 \mathit{XB}_1 = C_1$$, $$A_2 \mathit{XB}_2 = C_2$$, Proc. Cambridge philos. soc., 74, 213-216, (1973)  Morris, G.R.; Odell, P.L., Common solutions for n matrix equations with applications, J. assoc. comput. Mach., 15, 272-274, (1968) · Zbl 0157.22602  Peng, Y.X.; Hu, X.Y.; Zhang, L., An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation $$\mathit{AXB} = C$$, Appl. math. comput., 160, 763-777, (2005) · Zbl 1068.65056  Peng, Z.Y., An iterative method for the least squares symmetric solution of the matrix equation $$\mathit{AXB} = C$$, Appl. math. comput., 170, 711-723, (2005) · Zbl 1081.65039  G.X. Huang, F. Yin, Constrained inverse eigenproblem and associated approximation problem for anti-Hermitian R-symmetric matrices, Appl. Math. Comput. (2006), doi: 10.1016/j.amc.2006.07.11. · Zbl 1116.65047  G.-X. Huang, F. Yin, Matrix inverse problem and its optimal approximation problem for R-symmetric matrices, Appl. Math. Comput. (2006), doi: 10.1016/j.amc.2006.11.157.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.