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 $X^{\mathrm{T}}D$ 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 tr$(X^{\mathrm{T}}D)$. This paper is concerned with bounding the change in orthonormal basis matrix $X$ as subspace $\mathcal{X}$ varies under the requirement that $X^{\mathrm{T}}D$ 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.