Characterizing Positive Definite Matrices with t-norms

摘要

In this work symmetric positive definite matrices with non-negative entries will be characterized by using the Yager's family of t-norms. It can always be assumed that such an n × n matrix corresponds to a reflexive and symmetric fuzzy relation A on a set of cardinality n and then A is positive definite if and only if it is transitive with respect to a specific t-norm T_λ of Yager's family with λ depending on n. The result will be applied to give alternative proofs of the following two facts. Every min-indistinguishability operator separating points on a finite set has a positive definite matrix. Every ultrametric on a finite set is Euclidean.