an:03973066
Zbl 0603.15006
Sierksma, Gerard; Bakker, Evert Jan
Nearly principal minors of M-matrices
EN
Compos. Math. 59, 73-79 (1986).
00152120
1986
j
15B48
irreducible nonsingular M-matrix; nearly principal minors; Perron- Frobenius eigenvalue; Metzler's theorem
Let \(c_{ij}\) be the cofactor of the (i,j) element of the \(n\times n\) matrix (sI-A) where \(A\geq 0\), \(s>0\), and the Perron-Frobenius eigenvalue of A is \(<s\). Then it is known [the first author, Linear Algebra Appl. 26, 175-201 (1979; Zbl 0409.90027)] that when \(n\geq 3\), \(c_{ii}c_{kj}-c_{ij}c_{ki}\geq 0\). If, more stringently, \(A>0\) and all row sums of A are strictly less than s, then \(c_{kk}>c_{kj}\), \(j\neq k\) ('Metzler's theorem'). Both these propositions are generalized here, the first to larger 'nearly principal' minors of the matrix \(\{c_{ij}\}\). In the second the conditions on A are relaxed; unmentioned is information in the paper of \textit{T. Fujimoto}, \textit{C. Herrero} and \textit{A. Villar} [ibid. 64, 85-91 (1985; Zbl 0556.15003)] including the generalization of Metzler's theorem by \textit{M. Fiedler} and \textit{V. Pt??k} [Czech. Math. J. 12, 382-400 (1962; Zbl 0131.248)].
Eugene Seneta (Sydney)
Zbl 0409.90027; Zbl 0556.15003; Zbl 0131.248