关于符号模式矩阵允许对角化的新结果

New Results on Sign Patterns that Allow Diagonalizability

中文关键词:  符号模式矩阵  允许对角化  最大的环长  最小秩  最大秩  Frobenius 标准形

英文关键词:sign pattern  allowing diagonalizability  maximum cycle length  minimum rank  maximum rank  Frobenius normal form

中文摘要:

      刻画符号模式矩阵允许对角化一直是一个长期的开问题.在本文中,我们获得了一些符号模式矩阵允许对角化的充分和(或)必要条件.另外,我们还给出了从一个秩为${\rm mr}(A)$的实矩阵$B\in Q(A)$出发需要改变多少个元素能够获得一个秩为${\rm MR}(A)$的实矩阵$B'\in Q(A)$ 的一些结论.最后,我们还获得了一些用Frobenius标准形表示的符号模式矩阵的允许对角化的结论.

英文摘要:

      Characterization of sign patterns that allow diagonalizability has been a long-standing open problem. In this paper, we obtain some sufficient and/or necessary conditions for a sign pattern to allow diagonalizability. Moreover, we determine how many entries need to be changed to obtain a matrix $B'\in Q(A)$ with rank ${\rm MR}(A)$ from a matrix $B \in Q(A)$ with rank ${\rm mr}(A)$. Finally, we also obtain some results on a sign pattern matrix in Frobenius normal form that allows diagonalizability.