论文标题
投影且弱同时可对角矩阵及其应用
Projectively and weakly simultaneously diagonalizable matrices and their applications
论文作者
论文摘要
同时表征可对角（SD）矩阵的表征在最近几十年中，由于其广泛的应用及其在矩阵分析中的作用，人们一直受到极大的关注。但是，可以说，SD矩阵的概念仍然限制在更广泛的应用中。在本文中，我们考虑了与矩阵同时对角线相关的两项错误度量，并提出了几种SD的新变体。特别是TWSD，TWSD-B，T_ {M，N} -SD（SDO），DWSD和D_ {M，N} -SD（SDO）。这些都是SD的弱形式。我们在不同的假设下得出了它们的各种足够和/或必要条件，并显示了这些新概念之间的关系。最后，我们讨论了这些新概念的应用，例如，二次约束二次编程（QCQP）和独立的组件分析（ICA）。
Characterizing simultaneously diagonalizable (SD) matrices has been receiving considerable attention in the recent decades due to its wide applications and its role in matrix analysis. However, the notion of SD matrices is arguably still restrictive for wider applications. In this paper, we consider two error measures related to the simultaneous diagonalization of matrices, and propose several new variants of SD thereof; in particular, TWSD, TWSD-B, T_{m,n}-SD (SDO), DWSD and D_{m,n}-SD (SDO). Those are all weaker forms of SD. We derive various sufficient and/or necessary conditions of them under different assumptions, and show the relationships between these new notions. Finally, we discuss the applications of these new notions in, e.g., quadratically constrained quadratic programming (QCQP) and independent component analysis (ICA).
