A sensitivity analysis for linear systems involving M-matrices and its application to the 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.
scientific article; zbMATH DE number 3886057
scientific article