Brauer定理与Shemesh定理的改进

AN IMPROVEMENT OF BRAUER'S THEOREM AND SHEMESH'S THEOREM

本文引进了矩阵有向图的κ-path覆盖概念,借此给出一个新的特征值分布定理,改进了经典的Brauer定理;进而给出一类矩阵的秩的下界,改进了Shemes定理.

In this paper, we introduce the concept of κ-path cover about the directed graph of matrix. With the help of the κ-path cover we obtain a new distribution theorem on eigenvalues of matrix, and improve the classical Brauer's theorem. Moreover, we obtain lower bounds for the rank of some matrices, and improve Shemesh's theorem.