On the maximum rank of totally nonnegative matrices

Let A∈Rn×n be a totally nonnegative matrix with principal rank p, that is, every minor of A is nonnegative and p is the size of the largest invertible principal submatrix of A. We introduce the sequence of the first p-indices of A as the first initial row and column indices of a p×p invertible principal submatrix of A with rank p. Then, we study the linear dependence relations between the rows and columns indexed by the sequence of the first p-indices of A and the remaining of its rows and columns. These relations, together with the irreducibility property of some submatrices of A, allow us to present an algorithm that calculates the maximum rank of A as a function of the distribution of the first p-indices. Finally, we present a method to construct n×n totally nonnegative matrices with given rank r, principal rank p and a specific sequence of the first p-indices.