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Nearly principal minors of M-matrices. (English) Zbl 0603.15006
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 T. Fujimoto, C. Herrero and A. Villar [ibid. 64, 85-91 (1985; Zbl 0556.15003)] including the generalization of Metzler’s theorem by M. Fiedler and V. Pták [Czech. Math. J. 12, 382-400 (1962; Zbl 0131.248)].
15B48 Positive matrices and their generalizations; cones of matrices
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