矩阵的秩和非零特征值个数的关系研究

Discussions on the Relationship Between Ranks and Numbers of Non-zero Eigenvalues of Matrices

矩阵的秩和非零特征值个数是矩阵的重要不变量,研究二者关系也成为线性代数一个基本的问题.已有的文献分别给出了n阶矩阵的秩和非零特征值个数相等或相差n-1的充要条件.而矩阵指数又是矩阵的重要不变量,对复矩阵而言它指矩阵零特征值约当块的最大阶数.在已有文献基础上,研究了复数域上矩阵的秩和非零特征值个数二者的差与矩阵指数的关系,得到了矩阵的秩和非零特征值个数的差用矩阵指数刻画的一个充分必要条件,推广了已有文献的结果.

The rank and the number of non-zero eigenvalues of a matrix are two important invariants and the relation between these two values is a basic problem in the linear algebra. Some authors have described the necessary and sufficient conditions for that the rank and the number of non-zero eigenvalues are equal or have a gap of n-1. On the other side, the index of matrix is another important invariant, which, roughly speaking, is the maximal size of the zero eigenvalues in the canonical form of a complex matrix. Based on the existing research results, the relation of the gap between the rank and the number of non-zero eigenvalues with the index of matrix was investigated, and the necessary and sufficient conditions for these invariants were obtained, which is a generalization of some known results.