Abstract It is known that for each k ≥ 4 , there exists an irreducible sign pattern with minimum rank k that does not allow diagonalizability. However, it is shown in this paper that every square sign pattern A with minimum rank 2 that has no zero line allows diagonalizability with rank 2 and also with rank equal to the maximum rank of the sign pattern. In particular, every irreducible sign pattern with minimum rank 2 allows diagonalizability. On the other hand, an example is given to show the existence of a square sign pattern with minimum rank 3 and no zero line that does not allow diagonalizability; however, the case for irreducible sign patterns with minimum rank 3 remains open. In addition, for a sign pattern that allows diagonalizability, the possible ranks of the diagonalizable real matrices with the specified sign pattern are shown to be lengths of certain composite cycles. Some results on sign patterns with minimum rank 2 are extended to sign pattern matrices whose maximal zero submatrices are “strongly disjoint” (that is, their row index sets as well as their column index sets are pairwise disjoint).

中文翻译：

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允许对角化的符号模式的等级条件

摘要 众所周知，对于每个 k ≥ 4 ，都存在一个不可约符号模式，其最小秩 k 不允许对角化。然而，本文表明，每个最小秩为 2 且没有零线的方形符号模式 A 允许具有秩 2 并且秩等于符号模式的最大秩的对角化。特别是，每个最小秩为 2 的不可约符号模式都允许对角化。另一方面，举例说明了最小秩为 3 且没有不允许对角化的零线的方形符号模式的存在；然而，对于具有最小等级 3 的不可约符号模式的案例仍然开放。此外，对于允许对角化的符号模式，具有指定符号模式的可对角化实矩阵的可能秩显示为某些复合循环的长度。最小秩为 2 的符号模式的一些结果被扩展到最大零子矩阵“强不相交”的符号模式矩阵（即，它们的行索引集以及它们的列索引集是成对不相交的）。