Variations of orthonormal basis matrices of subspaces

An orthonormal basis matrix $ X $ of a subspace $ {\mathcal X} $ is known not to be unique, unless there are some kinds of normalization requirements. One of them is to require that $ X^{ \text{T}}D $ is positive semi-definite, where $ D $ is a constant matrix of apt size. It is a natural one in multi-view subspace learning models in which $ X $ serves as a projection matrix and is determined by a maximization problem over the Stiefel manifold whose objective function contains and increases with $ \text{tr}(X^{ \text{T}}D) $. This paper is concerned with bounding the change in orthonormal basis matrix $ X $ as subspace $ {\mathcal X} $ varies under the requirement that $ X^{ \text{T}}D $ stays positive semi-definite. The results are useful in convergence analysis of the NEPv approach (nonlinear eigenvalue problem with eigenvector dependency) to solve the maximization problem.