On the Explicit Expression of Chordal Metric between Generalized Singular Values of Grassmann Matrix Pairs with Applications
Weiwei Xu, Hong‐Kui Pang, Wen Li, Xueping Huang, Wenjie Guo · SIAM Journal on Matrix Analysis and Applications · 2018
In this paper we provide the explicit expression and sharper bounds of the chordal metric between generalized singular values of Grassmann matrix pairs. The new results involve the constrained optimization problem of the form $\max_{U\in\mathbb{U}_n} f(U)$, where $\mathbb{U}_n$ is the set of $n\times n$ unitary matrices. The optimal solution of this constrained optimization problem is also given. The proposed results can be applied to comparative analysis of gene mRNA expression data for mice macrophage under different conditions in actual experiments on mice. Some practical numerical examples in the data analysis of a gene mRNA expression level of a mice macrophage experiment are given to illustrate the efficiency of the theoretical results.