Nonsingular almost strictly sign regular matrices (Q414700)

The aim of the paper is to introduce and to study the class of almost strictly sign regular matrices.NEWLINENEWLINEThe set of strictly increasing sequences of \(k\) natural numbers, \(1\leq k\leq n\) is denoted by \(Q_{k,n}\). For a matrix \(A\in{\mathbb R}^{n\times n}\) and \(\alpha,\beta\in Q_{k,n}\), \(A[\alpha,\beta]\) denotes the submatrix of \(A\) with rows and columns indexed by \(\alpha\) and \(\beta\) respectively. A \textit{signature} is a real sequence \(\epsilon=(\epsilon_1,\dots,\epsilon_n)\) with \(|\epsilon_i|=1,\;i=1,\dots,n\). A minor of \(A\) is a \textit{trivial minor} if it is zero exactly according to the zero-nonzero pattern.NEWLINENEWLINEIf for all nontrivial minors \(\epsilon_k\det A[\alpha,\beta]>0\) for all \(\alpha,\beta\in Q_{k,n}\) with any \(k\), \(1\leq k\leq n\), then \(A\) is called \textit{almost strictly sign regular matrix with signature} \(\epsilon\).NEWLINENEWLINEThe class of almost strictly sign regular matrices includes the class of almost strictly totally positive matrices, defined by \(\det A[\alpha,\beta]>0\) for all \(\alpha,\beta\in Q_{k,n}\) with any \(k\), \(1\leq k\leq n\).NEWLINENEWLINESome characterizations of nonsingular almost strictly sign regular matrices are provided in terms of their nontrivial minors using consecutive rows and consecutive columns or in terms of the signs of boundary almost trivial minors.