Spectrally arbitrary patterns
An \(n\times n\) 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 \(n \times n\) 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 \(n_3\in \{0,1,2, n-1,n\}\) for all \(n\geq 2\). This leads to a proof of the conjecture for \(n\leq 5\). The truth of the conjecture is extended to \(n\leq 7\) by showing the stronger result that \(T_n\) is a SAP. Further questions about patterns that are SAPs and IAPs are considered.
scientific article; zbMATH DE number 1446347