Scalar Multiplication and Related Measures for Interval-Valued Intuitionistic Fuzzy Matrices

This is the authors' abstract. We don't add key points for this paper.

In this paper, we study several measures and algebraic properties of interval-valued intuitionistic fuzzy matrices. We define the Hamming, generalized Hamming, Euclidean, and generalized Euclidean measures and prove their main properties. We also show that these measures remain unchanged when both matrices are transposed. Next, we introduce scalar multiplication using a pair of parameters and examine its associativity and distributive properties. We then establish several identities and inequalities relating this operation to the Hamming and Euclidean measures. Examples illustrate how scalar multiplication works and help explain the theoretical results. These findings extend the algebraic framework of interval-valued intuitionistic fuzzy matrices and offer a starting point for studying further operations and their relationships with matrix measures.