Nested sequences of principal minors and potential stability
An \(n\times n\) sign-pattern \(A=[a_{ij}]\) with \(a_{ij}\in \{+,0,- \}\) defines a sign-pattern class \(Q(A)\), consisting of all real \(n\times n\) matrices \(B=[b_{ij}]\) with \(\text{sign } (b_{ij})=a_{ij}\). A sign-pattern \(A\) is said to be stable if all matrices \(B\in Q(A)\) are stable (i.e. the eigenvalues of \(B\) have negative real parts), and potentially stable if there exists a stable \(B\in Q(A)\). If \((\alpha_1,\dots,\alpha_n)\) is a permutation of \((1,\dots,n)\) let \(B(\alpha_1,\dots,\alpha_k)\) be the submatrix of \(B\), corresponding to rows and columns \(\alpha_1,\dots,\alpha_k\). A sign-pattern \(A\) allows a nested sequence of properly signed principal minors if there exist \(B\in Q(A)\) and \((\alpha_1,\dots,\alpha_n)\) such that \(\text{sign(det} (B(\alpha_1,\dots,\alpha_k)))= (-1)^k\) for \(k=1,\dots,n\). In this paper the authors give sufficient conditions for a sign-pattern \(A\) to allow a nested sequence of non-zero principal minors. A method is given to \(\text{sign }A\) so that it allows a nested sequence of properly signed principal minors. In this case, according to a classical result of \textit{M. E. Fisher} and \textit{A. T. Fuller} [Proc. Cambridge Philos. Soc. 54, 417-425 (1958; Zbl 0085.33102)] and \textit{C. S. Ballantine} [Proc. Am. Math. Soc. 25, 728-734 (1970; Zbl 0228.15001)] the sign-pattern \(A\) is potentially stable. Necessary and sufficient conditions for a certain tree sign-pattern to be potentially stable are also given.
scientific article; zbMATH DE number 1040644