Completion of an orthogonal projector and a related inverse mapping problem

We study the completion problem for an orthogonal projector A of Rn 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.