Multiplicative principal-minor inequalities for tridiagonal sign-symmetric \(P\)-matrices (Q5950102)

Three classes of matrices (positive definite, invertible totally nonnegative, and \(M\)-matrices) are considered and the problem is studied which of the ratios of products of principal minors are bounded over all matrices in the given class. This problem is solved for the class of tridiagonal sign-symmetric \(P\)-matrices. This class lies essentially in each of the three classes. The main problem is to characterize (via set-theoretic conditions) all pairs of collections of index sets \(\alpha, \beta\) such that \(\alpha(A)/ \beta (A)\leq K\) for some constant \(K\geq 0\).