Some results on a partition of \(Z\)-matrices
A \(Z\)-matrix is a real matrix in which each off-diagonal entry is nonpositive. For each nonnegative \(n \times n\) matrix \(B\) and \(k = 1, \ldots, n\), we denote by \(\rho_k (B)\) the maximum of the spectral radii of the principal \(k \times k\) submatrices of \(B\); and we put \(\rho_0 (B) = - \infty\) and \(\rho_{n + 1} (B) = \infty\). \textit{M. Fiedler} and \textit{T. L. Markham} [Linear Algebra Appl. 173, 115-124 (1992; Zbl 0754.15016)] considered the classes \(L_k\) \((k = 0, \ldots, n)\) consisting of real \(n \times n\) matrices of the form \(tI - B\) with \(B\) nonnegative and \(\rho_k (B) \leq t < \rho_{k + 1} (B)\). They showed that every \(n \times n\) \(Z\)-matrix lies in exactly one of these classes. The present paper considers properties of \(L_k\)-matrices and their Schur complements. A typical result is as follows (Theorem 2.1): An \(n \times n\) \(Z\)-matrix \(A\) is an \(L_k\)-matrix if and only if \(A^{-1}\) has a principal minor of order \(n - k - 1\) which is opposite in sign to \(\text{det} A\), but no principal minor of \(A^{-1}\) of larger order has this property.
scientific article; zbMATH DE number 778277
scientific article