# zbMATH — the first resource for mathematics

The skew-symmetric orthogonal solutions of the matrix equation $$AX=B$$. (English) Zbl 1128.15301
Summary: An $$n \times n$$ real matrix $$X$$ is said to be a skew-symmetric orthogonal matrix if $$X^{\text T} = -X$$ and $$X^{\text T}X = I$$. Using the special form of the C-S decomposition of an orthogonal matrix with skew-symmetric $$k \times k$$ leading principal submatrix, this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the skew-symmetric orthogonal solutions of the matrix equation $$AX = B$$. In addition, in corresponding solution set of the equation, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm have been provided. Furthermore, the Procrustes problem of skew-symmetric orthogonal matrices is considered and the formula solutions are provided. Finally an algorithm is proposed for solving the first and third problems. Numerical experiments show that it is feasible.

##### MSC:
 15A24 Matrix equations and identities
Full Text:
##### References:
  Vetter, W.J., Vector structures and solutions of linear matrix equations, Linear algebra appl., 10, 181-188, (1975) · Zbl 0307.15003  Magnus, J.R.; Neudecker, H., The commutation matrix: some properties and applications, Ann. statist., 7, 381-394, (1979) · Zbl 0414.62040  Magnus, J.R.; Neudecker, H., The elimination matrix: some lemmas and applications, SIAM J. algebr. discrete methods, 1, 422-449, (1980) · Zbl 0497.15014  Don, F.J.H., On the symmetric solutions of a linear matrix equation, Linear algebra appl., 93, 1-7, (1987) · Zbl 0622.15001  Dai, H., On the symmetric solutions of linear matrix equations, Linear algebra appl., 131, 1-7, (1990) · Zbl 0712.15009  Heney, E.W.C., Introduction to approximation theory, (1966), McGraw-Hill New York  Xie, D.; Hu, X.; Zhang, L., The solvability conditions for inverse eigenvalue problem of anti-bisymmetric matrices, J. comput. math. (China), 20, 3, 245-256, (2002) · Zbl 1005.65036  Zhang, L.; Xie, D., The least-square solution of subspace revolution, J. hunan univ., 19, 1, 115-120, (1992) · Zbl 0762.65024  Zhang, Z.; Hu, X.; Zhang, L., The solvability for the inverse eigenvalue problem of Hermitian-generalized Hamiltonian matrices, Inverse problems, 18, 1369-1376, (2002) · Zbl 1014.65030  Xie, D.; Hu, X.; Zhang, L., 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)  Schonemann, P.H., A generalized solution of the orthogonal procrustes problem, Psychomctrika, 31, 1-10, (1966) · Zbl 0147.19401  Higham, N.J., The symmetric procrustes problem, Bit, 28, 133-143, (1988) · Zbl 0641.65034  Golub, G.H.; Van Loan, C.F., Matrix computation, (1996), The Johns Hopkins University Press USA
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.