Multiplicative principal-minor inequalities for totally nonnegative matrices
An \(n\times n\) matrix \(A\) is called totally nonnegative (TN) if every minor of \(A\) is nonnegative. Several inequalities are known among products of principal minors, especially for positive definite and TN matrices. Examples of these inequalities are the Hadamard, Fischer and Koteljanskii inequalities. The paper under review is motivated by the question of finding the set of possible inequalities involving determinants (i.e. inequalities among principal minors) for a given class of matrices, and in particular for the class of TN matrices. The authors succeed in identifying all possible inequalities among products of principal minors for invertible TN matrices (and thus, for TN matrices by a density argument) when \(n\leq 5\), and they characterize several cases for general \(n\). In all of the cases studied by the authors the constant in the inequality is always one, and the authors conjecture that this is true in general. The characterization of the inequalities is done in terms of combinatorial objects.
scientific article; zbMATH DE number 1981406
scientific article