On characterizing Z-matrices (Q5955650)

The class of \(n\times n\) \(Z\)-matrices (i.e. matrices with nonpositive off-diagonal entries) is considered, as well as the Fiedler-Markham partition \(Z=\{{\mathcal L}_k\}\) of this class; \({\mathcal L}_k (1\leq k\leq n)\) is the set of matrices \(A\) of the form \(A=tI-B\) in which \(B\geq 0\) and \(\rho_k(B)\leq t< \rho_{k+1}(B)\), where \(\rho_k(B)\), \(1\leq k\leq n\), denotes the maximum of the spectral radii of all \(k\times k\) principal submatrices of \(B\), \(\rho_{n+1} (B)= \infty\) and \({\mathcal L}_0\) is the set of matrices \(A=tI-B\) with \(t<\rho_1 (B)\).NEWLINENEWLINENEWLINETransformation characterizations are given for \(Z\)-matrices and for \({\mathcal L}_k\)-matrices as well. For instance, it is shown (Theorem 1.3) that \(A\in {\mathcal L}_k\) if and only if (i) for each \(C\neq x\in\mathbb{R}^k\) and every \(\alpha\subset \{1,\dots, n\}\) with \(|\alpha |=k\), \(x\) and \(A[\alpha]x\) are doubly closed sign-related and (ii) there is \(\beta\subset \{1,\dots,n\}\) with \(|\beta|=k+1\) and a vector \(\widehat x\), \(0<\widehat x\in \mathbb{R}^{k+1}\) such that \(A[\beta] \widehat x<0\). Here \(A[\alpha]\) denotes the submatrix of \(A\) containing the rows and columns indexed by \(\alpha\subset \{1,\dots,n\}\).NEWLINENEWLINENEWLINEThese characterizations generalize the transformational characterization of nonsingular \(M\)-matrices given by \textit{C. R. Johnson} and the author [ibid. 330, No. 1-3, 43-48 (2001; Zbl 0985.15022)].