Derivatives of the Perron root of a positive reciprocal matrix: with application to the analytic hierarchy process. (English) Zbl 0619.15017

This paper presents a refinement of the classical results on the derivatives of the Perron root of an essentially nonnegative matrix to a special class of essentially nonnegative matrices, namely, positive reciprocal matrices. This class of matrices arises in Saaty’s analytic hierarchy process, a decision-analysis method which is based upon pairwise comparisons among a set of alternatives over various criteria. Formulas for the first and second partial derivatives of the Perron root of a positive reciprocal matrix are developed, and the use of these formulas in conjunction with Saaty’s method is illustrated.
Reviewer: S.L.Campbell


15B48 Positive matrices and their generalizations; cones of matrices
15A18 Eigenvalues, singular values, and eigenvectors
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