论文标题
通过遗传学矩阵的一致性同时对角线：一些等效条件和数值解决方案
Simultaneous diagonalization via congruence of Hermitian matrices: some equivalent conditions and a numerical solution
论文作者
论文摘要
本文旨在解决Hermitian SDC问题，即\ textit {同时通过$*$ - 一致性对角度化}，这是有限的许多（不需要成对通勤）遗产矩阵的集合。从理论上讲，我们提供了一些等效条件，因为可以通过$^*$ - 一致性同时对这种矩阵集合进行对角线。％通过非词组矩阵。有趣的是，这样的条件之一导致对半决赛（SDP）的积极确定解决方案。从实际的角度来看，我们提出了一种用于解决此类问题的算法。提出的算法是 （1）一个积极的半决赛计划，检测最初的遗产矩阵是否可以通过$*$ - 一致性同时对角线，并且 （2）一种类似雅各比的算法，用于通过$*$同时对角线化的算法 - 一致性的通勤正常矩阵从上一个阶段得出。还提供了用手/编码的示例\ textsc {matlab}中的示例。
This paper aims at solving the Hermitian SDC problem, i.e., that of \textit{simultaneously diagonalizing via $*$-congruence} a collection of finitely many (not need pairwise commute) Hermitian matrices. Theoretically, we provide some equivalent conditions for that such a matrix collection can be simultaneously diagonalized via $^*$-congruence.% by a nonsingular matrix. Interestingly, one of such conditions leads to the existence of a positive definite solution to a semidefinite program (SDP). From practical point of view, we propose an algorithm for numerically solving such problem. The proposed algorithm is a combination of (1) a positive semidefinite program detecting whether the initial Hermitian matrices are simultaneously diagonalizable via $*$-congruence, and (2) a Jacobi-like algorithm for simultaneously diagonalizing via $*$-congruence the commuting normal matrices derived from the previous stage. Illustrating examples by hand/coding in \textsc{Matlab} are also presented.
