Inertially arbitrary sign patterns with no nilpotent realization

Abstract

An n by n sign pattern S is inertially arbitrary if each ordered triple ( n 1, n 2, n 3) of nonnegative integers with n 1 + n 2 + n 3 = n is the inertia of some real matrix in Q ( S ) , the sign pattern class of S . If every real, monic polynomial of degree n having a positive coefficient of x n−2 is the characteristic polynomial of some matrix in Q ( S ) , then it is shown that S is inertially arbitrary. A new family of irreducible sign patterns G 2 k + 1 ( k ⩾ 2 ) is presented and proved to be inertially arbitrary, but not potentially nilpotent (and thus not spectrally arbitrary). The well-known Nilpotent-Jacobian method cannot be used to prove that G 2 k + 1 is inertially arbitrary, since G 2 k + 1 has no nilpotent realization. In order to prove that Q ( G 2 k + 1 ) allows each inertia with n 3 ⩾ 1, a realization of G 2 k + 1 with only zero eigenvalues except for a conjugate pair of pure imaginary eigenvalues is identified and used with the Implicit Function Theorem. Matrices in Q ( G 2 k + 1 ) with inertias having n 3 = 0 are constructed by a recursive procedure from those of lower order. Some properties of the coefficients of the characteristic polynomial of an arbitrary matrix having certain fixed inertias are derived, and are used to show that G 5 and G 7 are minimal inertially arbitrary sign patterns.