Inertially arbitrary (\(2r-1\))-diagonal sign patterns
The inertia of a matrix \(A\) is the triple (\(i_+(A)\), \(i_-(A)\), \(i_0(A)\)), where \(i_+\), \(i_-\) and \(i_0\) are the numbers of eigenvalues with positive, negative and zero real parts, respectively. An \(n\times n\) sign pattern \(S= (s_{ij})\) has \(s_{ij}\in \{+,-,0\}\) and the qualitative class of \(S\) is \(Q(S)= \{A= (a_{ij})\in M_n(R): \text{sign}(a_{ij})= s_{ij}\) for all \(i\), \(j\}\). The inertia of \(S\) is the set of triples \(i(S)= \{i(A): A\in Q(S)\}\). An \(n\times n\) sign pattern \(S\) is an inertially arbitrary pattern if \((n_1,n_2,n_3)\in i(S)\) for all nonnegative triples \((n_1,n_2,n_3)\) with \(n_1+ n_2+ n_3= n\). Consider the \(n\times n(2r-1)\)-diagonal sign pattern \(S_{n,r}\) with positive entry \((i,j)\) for \(1\leq j-i\leq r-1\) or \(i= j= n\), negative entry \((i,j)\) for \(1\leq i- j\leq r-1\) or \(i= j=1\), and zero entry otherwise. Extending previous results in the literature, it is proved that \((n-k,k,0)\in i(S_{n,r})\) for \(0\leq k\leq n\), \(2\leq r\leq n\) and that \(S_{n,n-1}\) is an inertially arbitrary pattern for \(n\geq 3\).
scientific article; zbMATH DE number 1864773