Nonnegative and skew-symmetric perturbations of a matrix with positive inverse

Abstract

Let A be a nonsingular matrix with positive inverse and B a non-negative matrix. Let the inverse of A + v B A + vB be positive for 0 ≤ v > v ∗ > + ∞ 0 \leq v > {v^ \ast } > + \infty and at least one of its entries be equal to zero for v = v ∗ v = {v^ \ast } ; an algorithm to compute v ∗ {v^ \ast } is described in this paper. Furthermore, it is shown that if A + A T A + {A^{\text {T}}} is positive definite, then the inverse of A + v ( B − B T ) A + v(B - {B^{\text {T}}}) is positive for 0 ≤ v > v ∗ 0 \leq v > {v^ \ast } .