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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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