an:03886057 Zbl 0556.15003 Fujimoto, Takao; Herrero, Carmen; Villar, Antonio A sensitivity analysis for linear systems involving M-matrices and its application to the Leontief model EN Linear Algebra Appl. 64, 85-91 (1985). 00149858 1985
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15A06 93B35 15B48 91B60 sensitivity analysis; M-matrices; Metzler theorem; Morishima theorem; Leontief model Let $$M^ 0$$ and $$M^*$$ be (n$$\times n)$$ non-singular M-matrices (n$$\geq 2)$$. (That is: $$M^ 0$$ and $$M^*$$ may be written as $$M^ 0=\lambda I- A^ 0$$, $$M^*=\lambda I-A^*$$ where $$\lambda$$ is a positive scalar, and $$A^ 0$$ and $$A^*$$ are non-negative matrices, and $$(M^ 0)^{- 1}$$, $$(M^*)^{-1}$$ are also non-negative.) $$M^ 0$$ and $$M^*$$ may differ only in the first s $$(0<s<n)$$ columns. Let $$w^ 0$$ and $$w^*$$ be strictly positive n-vectors, which coincide in their last (n-s) entries. Let $$p^ 0=(M^ 0)^{-1}w^ 0$$, $$p^*=(M^*)^{-1}w^*$$. Then if $$(p^ 0M^*)_ i>w^*_ i$$ for $$i\in S$$, the authors prove $$\min_{i\in S}\{p^*_ i/p^ 0_ i\}<\min_{i\in R}\{p^*_ i/p^ 0_ i\}$$, where $$S=\{1,2,...,s\}$$, $$R=\{s+1,...,n\}$$. This is a partial generalization of Theorem 21 of \textit{G. Sierksma} [Linear Algebra Appl. 26, 175-201 (1979; Zbl 0409.90027)]; see also the reviewer's paper [Non-negative matrices and Markov chains (1981; Zbl 0471.60001 pp. 35- 39], in that changes in the M-matrix $$\{$$ from $$M^ 0$$ to $$M^*\}$$ are also permitted. Eugene Seneta (Sydney) Zbl 0409.90027; Zbl 0471.60001