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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)].
##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices
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##### References:
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