Computing the inertia from sign patterns

A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang showed that the inertia of a sign-nonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such matrices. The algorithm runs in O(nm) time for a symmetric matrix of order n with m nonzero entries. The correctness of the algorithm provides an alternative proof of the result by Hall et al. In addition, for a symmetric matrix in general, it is shown to be NP-complete to decide whether the inertia of the matrix is not determined by the sign pattern.