Note on lower bounds for the rank of a matrix

The following results are proved: Let A = (a_ij) be an n × n complex matrix, n ? 2, and let k be a fixed integer, 1 ? k ? n ? 1.(1) If there exists a monotonic G-function f = (f₁,…,f_n) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have

$\text{Π}\text{i∈S}\text{f}{}_{\mathrm{i}}\text{(A)<}\text{Π}\text{i∈S}\text{|a}{}_{\mathrm{ii}}\text{|,}$

then the rank of A is ? n ? k + 1. (2) If A is irreducible and if there exists a G-function f = (f₁,…,f_n) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have

$\text{Π}\text{i∈S}\text{f}{}_{\mathrm{i}}\text{(A)<}\text{Π}\text{i∈S}\text{|a}{}_{\mathrm{ii}}\text{|,}$

then the rank of A is ? n ? k + 1 if k ? 2, n ? 3; it is ? n ? 1 if k = 1.