On the spectrum of pairwise comparison matrices Original Research Article

This paper deals with the spectrum of symmetrically reciprocal (SR) perturbations of transitive matrices of nonzero complex entries. By introducing the so-called pseudo-circulant matrices, the spectral decomposition of specific SR perturbations of n×n transitive matrices and that of arbitrary 3×3 SR matrices are provided. Closed forms are given for all eigenvalues and their associated eigenvectors of these SR matrices, also known as pairwise comparison matrices (PCMs). To improve the evaluations at a given PCM, as well as the priority ranking of the alternatives, these results are applied to the analytic hierarchy process (AHP) using positive SR matrices. Explicit intervals for the ranges of possible rank reversals between the alternatives are established. The spectrum of certain SR perturbations of transitive matrices with nonzero complex entries is also derived which can be applied in vehicle system dynamics. Three application examples are included as well.