TY - JOUR
T1 - Rank conditions for sign patterns that allow diagonalizability
AU - Feng, Xin Lei
AU - Gao, Wei
AU - Hall, Frank J.
AU - Jing, Guangming
AU - Li, Zhongshan
AU - Zagrodny, Chris
AU - Zhou, Jiang
N1 - Funding Information:
This research is supported in part by National Natural Science Foundation of China (11601102), China Scholarship Council (2014085151114), and Research Project of Leshan Normal University, PR China (Z1514, LZD016).
Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/5
Y1 - 2020/5
N2 - 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).
AB - 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).
KW - Allowing diagonalizability
KW - Composite cycles
KW - Maximum composite cycle length
KW - Minimum rank
KW - Rank-principal matrices
KW - Sign pattern
UR - http://www.scopus.com/inward/record.url?scp=85077929623&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85077929623&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2019.111798
DO - 10.1016/j.disc.2019.111798
M3 - Article
AN - SCOPUS:85077929623
SN - 0012-365X
VL - 343
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 5
M1 - 111798
ER -