#5765. Pairwise comparisons matrix decomposition into approximation and orthogonal component using Lie theory

July 2026publication date
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4 total number of authors per manuscript0 $

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Journal’s subject area:
Applied Mathematics;
Theoretical Computer Science;
Software;
Artificial Intelligence;
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Abstract:
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:
Approximate reasoning; Approximation; Consistency-driven; Decomposition; Inconsistency; Lie algebra; Matrix Lie group; Orthogonality; Pairwise comparison; Subjectivity

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