Recursive Multilevel Decomposition of Pairwise Comparison Matrices: Algorithms and Inconsistency Diagnostics
Positive reciprocal pairwise comparison matrices are studied in entrywise logarithmic coordinates. Starting from the established consistent-residual split, we show that, for any fixed ordering, the residual matrix admits a unique top-down decomposition into mutually Frobenius-orthogonal layers. The construction applies to every positive reciprocal PC matrix of size $n \geq 3$, without additional spectral, sparsity, or genericity assumptions. An explicit active-matrix recursion and an equivalent cumulative parameter-extraction procedure have computational complexities $O(n^3)$ and $O(n^2)$, respectively. The squared Frobenius norms of the layers define an ordering-conditioned residual-energy profile that provides structural information complementary to the geometric-mean priorities and the Koczkodaj inconsistency indicator (Kii). Entrywise exponentiation yields an exact Hadamard-product reconstruction of the original reciprocal matrix. Numerical experiments confirm agreement between the two computational procedures, exact reconstruction, mutual orthogonality, and numerical accuracy to machine precision.