The hadamard product of a positive reciprocal matrix and some results in AHP

We define four types of standard matrices, prove that every positive reciprocal matrix is the Hadamard product of one type of standard matrix and a corresponding consistent matrix, and conclude that the set of EM, LDM, LLSM, REM, and GEM perturbation matrices is equal to the set of corresponding type of standard matrices. We also provide a necessary and sufficient condition that the priority vectors of an arbitrary positive reciprocal matrix in EM, LDM, LLSM, REM, and GEM are equal. We conclude our paper by applying this decomposition to the analysis of sensitivity to compute the priority vector of a positive reciprocal matrix obtained from the Hadamard product of a perturbed matrix and a perturbation matrix.