Fuzzy Matrices

Like classical (crisp) matrix theory, fuzzy matrix (FM) is also a very useful tool for modelling many uncertain problems that arise in sciences, engineering, social sciences, and many other areas. In crisp matrices, the elements are either real numbers or complex numbers or sometimes vectors, but in FMs, the elements are membership values. In the Boolean matrix, the elements are either 0 or 1 and the two basic operations addition and multiplication are max and min, i.e. \(a+b=\max \{a,b\}\) and \(a.b=\min \{a,b\}\) . Here, 0 and 1 represent two states of a system, such as on and off of an electrical network, etc. Whereas in FM the elements are any real number in the closed interval \([0,1]\) , so it is a multi-state logic, i.e. it is used to represent infinite many situations. The addition and multiplication rules are the same as a Boolean matrix. Fuzzy matrices are used to model problems of many fields, e.g. fuzzy relations, fuzzy relational equations, pattern classification, knowledge-based systems, etc.