Abstract In this paper, we start from the premise that pairwise comparisons between alternatives can be modeled by means of the additive representation of preferences. In this setting we study some algebraic properties of three sets: the set of pairwise comparison matrices, its subset of consistent ones and the orthogonal complement of the latter. The three sets are all vector spaces and we propose and interpret simple bases for each one. We prove that a convenient inner product can be found in the three cases such that the corresponding basis is orthonormal with respect to the considered inner product. In addition (i) we prove that the well-known method of the logarithmic least squares used to estimate the weight vector can be reinterpreted by referring to a basis for the set of consistent preferences and (ii) we interpret a transformation recently proposed by Csato.

中文翻译：

---

成对比较的线性代数

摘要 在本文中，我们从可以通过偏好的加法表示对备选方案之间的成对比较进行建模的前提开始。在这种情况下，我们研究了三个集合的一些代数性质：成对比较矩阵的集合、其一致矩阵的子集以及后者的正交补集。这三个集合都是向量空间，我们为每个集合提出并解释了简单的基。我们证明了在三种情况下都可以找到一个方便的内积，使得相应的基础相对于所考虑的内积是正交的。