An orthonormal basis matrix $X$ of a subspace ${\cal X}$ is known not to be unique, unless there are some kinds of normalization requirements. One of them is to require that $X^{\rm T}D$ 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$(X^{\rm T}D)$. This paper is concerned with bounding the change in orthonormal basis matrix $X$ as subspace ${\cal X}$ varies under the requirement that $X^{\rm T}D$ 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.

翻译：子空间$\cal X$的正交基矩阵$X$已知并非唯一，除非施加某种归一化约束。其中一种约束要求$X^{\rm T}D$为半正定矩阵，其中$D$为尺寸适当的常数矩阵。该约束在多视角子空间学习模型中具有自然属性，此时$X$作为投影矩阵，通过斯提弗尔流形上的最大化问题确定，其目标函数包含且随tr$(X^{\rm T}D)$递增。本文研究在$X^{\rm T}D$保持半正定的约束下，正交基矩阵$X$随子空间$\cal X$变化时的变化界限。该结果对求解该最大化问题的NEPv方法（含特征向量依赖的非线性特征值问题）的收敛性分析具有实用价值。