论文标题
在半扇形矩阵的阶段
On the Phases of a Semi-Sectorial Matrix
论文作者
论文摘要
在本文中，我们将部门矩阵的阶段的定义扩展到了可能是单数的半扇形矩阵的定义。该阶段的性质也被扩展，包括摩尔 - 芬罗广义逆，压缩和Schur补充，矩阵总和和产品的性质。特别是，在$ ab $的非零特征值的阶段与$ a $ a $和$ b $的压缩阶段之间建立了多数关系，这导致了广义的矩阵小阶段定理。对于不一定是半扇区的矩阵，我们通过对角相似性转换来定义其（最大和最小）基本阶段。获得了有向图的拉普拉斯基质的必需阶段的显式表达。
In this paper, we extend the definition of phases of sectorial matrices to those of semi-sectorial matrices, which are possibly singular. Properties of the phases are also extended, including those of the Moore-Penrose generalized inverse, compressions and Schur complements, matrix sums and products. In particular, a majorization relation is established between the phases of the nonzero eigenvalues of $AB$ and the phases of the compressions of $A$ and $B$, which leads to a generalized matrix small phase theorem. For the matrices which are not necessarily semi-sectorial, we define their (largest and smallest) essential phases via diagonal similarity transformation. An explicit expression for the essential phases of a Laplacian matrix of a directed graph is obtained.
