The skew-symmetric orthogonal solutions of the matrix equation AX=B

An n × n real matrix X is said to be a skew-symmetric orthogonal matrix if XT = −X and XTX = I. Using the special form of the C-S decomposition of an orthogonal matrix with skew-symmetric k × 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.