Almost full rank matrices arising from transitive tournaments

Abstract

Let F be a field and suppose a := ( a 1 , a 2 , … ) is a sequence of non-zero elements in F . For a tournament T on [ n ] , associate the n × n symmetric matrix M T ( a ) (resp. skew-symmetric matrix M T , skew ( a ) ) with zero diagonal as follows: for i<j, if the edge ij is directed as i → j in T, then set M T ( a ) = a i (resp. M T , skew ( a ) = a i ), else set M T ( a ) = a j (resp. M T , skew ( a ) = a j ). Let M n ( a ) (resp. M n , skew ( a ) ) be the family consisting of all the n × n symmetric matrices M T ( a ) (resp. skew-symmetric matrices M T , skew ( a ) ) as T varies over all tournaments on [ n ] . We show that any matrix in M n ( a ) or M n , skew ( a ) corresponding to a transitive tournament has a rank at least n−1, and this is best possible. This settles in a strong form a conjecture posed in Balachandran et al. [An ensemble of high-rank matrices arising from tournaments; Linear Algebra Appl. 2023;658:310–318]. As a corollary, any matrix in these families has rank at least ⌊ log 2 ⁡ ( n ) ⌋ .