Spectrally arbitrary patterns

An n x n sign pattern matrix S is an inertially arbitrary pattern (IAP) if each nonnegative triple (n(1), n(2), n(3)) with n(1) + n(2) + n(3) = n is the inertia of a matrix with sign pattern S. Analogously, S is a spectrally arbitrary pattern (SAP) if, for any given real monic polynomial r(x) of order n, there is a matrix with sign pattern S and characteristic polynomial r(x). Focusing on tree sign patterns, consider the n x n tridiagonal sign pattern T-n with each superdiagonal entry positive, each subdiagonal entry negative, the (1, 1) entry negative, the (n, n) entry positive, and every other entry zero. It is conjectured that T-n is an IAP. By constructing matrices A(n) with pattern T-n, it is proved that T-n allows any inertia with n(3) is an element of {0, 1, 2, n -1, n} for all n greater than or equal to 2. This leads to a proof of the conjecture for n less than or equal to 5. The truth of the conjecture is extended to n less than or equal to 7 by showing the stronger result that T-n is a SAP. The proof of this latter statement involves finding a matrix A(n) with pattern T-n that is nilpotent. Further questions about patterns that are SAPs and IAPs are considered. (C) 2000 Elsevier Science Inc. All rights reserved.