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Abstract: We study the completion problem for an orthogonal projector A of from knowledge of one or several of its columns, as well as the related matrix inverse problem of finding an orthogonal projector A with AX =Y for two given n by k matrices X and Y. Using geometric properties of orthogonal projections and of the singular value decomposition, we derive theoretical and numerical solutions for these problems. We give algorithms and test their MATLAB implementations. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]