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).
翻译:近几十年来,由于SD矩阵的广泛应用及其在矩阵分析中的作用,同时可区分(SD)矩阵的特性在过去几十年中一直受到相当重视,然而,SD矩阵的概念对于更广泛的应用仍然可以说是限制性的。在本文中,我们考虑了与同时对矩阵进行对等化有关的两个错误措施,并提出了其中的若干新的SD变量:特别是TWSD、TWSD-B、T ⁇ m、n}SD(SDO)、DWSD和D ⁇ m、n}SD(SD) 。这些都是较弱的SD形式。我们在不同假设下得出了它们的各种充足和/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在诸如四面制约的四面形编程和独立构件分析(ICA)中的应用。