广义Jordan标准形与矩阵多项式的秩

摘要

通过刻画矩阵的初等因子对应的广义Jordan块的性质，在矩阵的初等因子组已知的前提下，给出了矩阵多项式秩的计算公式。作为结论的应用，考虑了当特征多项式和极小多项式相等时，矩阵多项式秩的情况。%In this paper, we describe the properties of the blocks of the generalized Jordan corresponding to the elementary factor of matrix. By using the properties, we give the computational formula of the rank of polynomial of matrix under the condition that all elementary factors of the given matrix are known. As an application, we consider the situation of the rank of polynomial of matrix when the characteristic polynomial is equal to the minimal polynomial.