发表机构
Yüksek İhtisas University; Laurentian University; Ankara University（Yüksek İhtisas大学; 劳伦森大学; 安卡拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出成对比较矩阵的递归多级正交分解方法，给出O(n^3)和O(n^2)两种算法，并利用残差能量剖面补充不一致性诊断信息。
AI 中文摘要
在逐元素对数坐标下研究正互反成对比较矩阵。从已确立的一致性-残差分解出发，我们证明，对于任意固定排序，残差矩阵具有唯一的自顶向下分解，分解为相互Frobenius正交的层。该构造适用于每个大小为 $n \geq 3$ 的正互反PC矩阵，无需额外的谱、稀疏性或一般性假设。一个显式的活动矩阵递归和一个等价的累积参数提取过程的计算复杂度分别为 $O(n^3)$ 和 $O(n^2)$。各层的Frobenius范数平方定义了一个排序条件下的残差能量剖面，该剖面提供了与几何平均优先级和Koczkodaj不一致性指标（Kii）互补的结构信息。逐元素指数化可精确重构原始互反矩阵的Hadamard积。数值实验证实了两种计算程序之间的一致性、精确重构、相互正交性以及达到机器精度的数值准确性。
英文摘要
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.
Comments24 pages, 1 figure, 4 tables
