对角占优矩阵奇异-非奇异的充分必要判据

Necessary and sufficient criterion for singularity or non-singularity of diagonally dominant matrices

本文研究对角占优矩阵奇异-非奇异的充分必要条件.基于Taussky定理,本文得出,可约对角占优矩阵的奇异性由其独立Frobenius块的奇异性决定,从而将这一问题化为不可约对角占优矩阵的奇异-非奇异性问题;运用Taussky定理研究奇异不可约对角占优矩阵的相似性和酉相似性,获得这类矩阵元素辐角间的关系;并与Taussky定理给出的这类矩阵元素模之间的关系结合在一起,研究不可约对角占优矩阵奇异的充分必要条件;最后给出不可约对角占优矩阵奇异-非奇异性的判定方法.

The necessary and sufficient conditions that a diagonally dominant matrix is singular or nonsingular are examined in this article. According to Taussky Theorem we find that the singularity of a reducible diagonally dominant matrix is determined by the singularity of its independent Frobenius blocks. Thus, whether a reducible diagonally dominant matrix is singular or not can be transformed into the problem of whether its Frobenius blocks, which are irreducible diagonally dominant matrices, are singular or nonsingular. According to Taussky Theorem we study the similarity and unitary similarity of the singular irreducible diagonally dominant matrices. Furthermore we obtain some relationship of arguments between the elements of this type of matrices. Incorporated with an existing relationship of modulus between the elements of this type of matrices given by Taussky, we study the necessary and sufficient conditions for singularity of this type of matrices. Finally we give the criteria for the singularity or non-singularity of the irreducible diagonally dominant matrices.