Spectrally arbitrary patterns Original Research Article

An n×n sign pattern matrix S is an inertially arbitrary pattern (IAP) if each nonnegative triple (n1,n2,n3) with n1+n2+n3=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×n tridiagonal sign pattern Tn 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 Tn is an IAP. By constructing matrices An with pattern Tn, it is proved that Tn allows any inertia with n3set membership, variant{0,1,2,n−1,n} for all ngreater-or-equal, slanted2. This leads to a proof of the conjecture for nless-than-or-equals, slant5. The truth of the conjecture is extended to nless-than-or-equals, slant7 by showing the stronger result that Tn is a SAP. The proof of this latter statement involves finding a matrix An with pattern Tn that is nilpotent. Further questions about patterns that are SAPs and IAPs are considered.