Inertially arbitrary sign patterns with no nilpotent realization Original Research Article

An n by n sign pattern image is inertially arbitrary if each ordered triple (n1, n2, n3) of nonnegative integers with n1 + n2 + n3 = n is the inertia of some real matrix in image, the sign pattern class of image. If every real, monic polynomial of degree n having a positive coefficient of xn−2 is the characteristic polynomial of some matrix in image, then it is shown that image is inertially arbitrary. A new family of irreducible sign patterns image 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 image is inertially arbitrary, since image has no nilpotent realization. In order to prove that image allows each inertia with n3 greater-or-equal, slanted 1, a realization of image with only zero eigenvalues except for a conjugate pair of pure imaginary eigenvalues is identified and used with the Implicit Function Theorem. Matrices in image with inertias having n3 = 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 image and image are minimal inertially arbitrary sign patterns.