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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
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