M-矩阵的判定

Judgment of M-matrix

M-矩阵是数值代数的一个重要研究课题。通过研究矩阵伴随有向图圈中所涉及到的量,得到了不可约矩阵是非奇异M-矩阵的一个新的充要条件,同时给出了一个将可约矩阵化为Frobenius标准型的图论方法,进而得到判定一个可约矩阵是否为非奇异M-矩阵的具体方法,即先将矩阵化为Frobenius标准型,然后判定对角线上各块是否是非奇异M-矩阵。最后通过一个实例说明所述的方法是可行的。

M-matrix is an important research task of numeric algebra. A new necessary and sufficient condition of non-singular M-matrix is obtained by researching quantity involved in circuit of associated digraph of matrix, in the meantime a method of transforming reducible matrix into Frobenius' standard form and judging M-property of reducible matrix is provided, i.e. firstly transforming matrix into Frobenius' standard form and then judging whether diagonal blocks belong to non-singular M-matrix or not. In the end, a practical example shows that the method is feasible.