  This paper examines the use of Lie group and Lie Algebra theory to construct
the geometry of pairwise comparisons matrices. The Hadamard product (also known
as coordinatewise, coordinate-wise, elementwise, or element-wise product) is
analyzed in the context of inconsistency and inaccuracy by the decomposition
method.
  The two designed components are the approximation and orthogonal components.
The decomposition constitutes the theoretical foundation for the multiplicative
pairwise comparisons.
  Keywords: AI, pairwise comparison; pairwise comparisons matrix, subjectivity,
matrix Lie group, Lie algebra, inconsistency; approximation, factorization,
orthogonality.
