On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices

Abstract

In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced.The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.

Constructive Consistent Approximations in Pairwise Comparisons

“…In the theory of pairwise comparisons there are considered two groups K and L in A: K of multiplicative type and L ⊂ A of additive type. It should be noticed that all pairwise comparisons problems are originally stated for the multiplicative group K. Since these problems are extremely hard to solve even by computers [1], [3], [4], [10] and [11], they are approximated by additive projections onto L-valued consistent matrices, which are eventually transformed by using abstract exponential functions. Definition 1.…”

“…e.g. [4], [10] and [1]) in an abstract setting, let K n×n be the set of all K-valued matrices M = [m ij ] n i.j=1 with entries in a multiplicative group K. Then K n×n is a multiplicative matrix group under the operation of pointwise multiplication of matrices M = [m ij ] and…”

“…According to [1,4,10,12] the classical pairwise comparisons theory is based on the approximation of inconsistent matrices by consistent projections. This approach can be extended to groups M n = M n (K) and A n = A n (L) of consistent matrices as specified below.…”

“…The consistent approximation plays fundamental role in the pairwise comparisons theory and its applications, cf. Saaty [1,2], Laarhoven and Pedrycz [3], Koczkodaj and Or lowski [4], Cavallo and Brunelli [5], Farkas et al [6], Koczkodaj and Szarek [7], Holsztyński and Koczkodaj [8], Koczkodaj and Szwarc [9], Koczkodaj et al [10], Smarzewski and Rutka [11] and other references therein. According to [1,3,4] a consistent projection is defined, for the multiplicative group of positive real and triangular fuzzy numbers, as a composition of additively and multiplicatively invariant matrix mappings.…”

Pairwise Comparisons Matrix Decomposition into Approximation and Orthogonal Component Using Lie Theory

“…Problem outline. In [16] (a follow-up to [19]), Smarzewski has observed that PC matrices form a group under Hadamard product. In this paper, we will show that it is, in fact, a Lie group.…”