Some extremal problems for strictly totally positive matrices

AbstractFor a strictly totally positive M × N matrix A we show that the ratio ∥Ax∥p∥x∥p has exactly R = min{ M, N} nonzero critical values for each fixed p ϵ (1, ∞). Letting λi denote the ith critical value, and xi an associated critical vector, we show that λ1 > … > λR > 0 and xi (unique up to multiplication by a constant) has exactly i − 1 sign changes. These critical values are generalizations to lp of the s-numbers of A and satisfy many of the same extremal properties enjoyed by the s-numbers, but with respect to the lp norm