A Comparative Review of the Fuzzy, Intuitionistic and Neutrosophic Numbers in Solving Uncertainty Matrix Equations

Matrix equations play a fundamental role in scientific and engineering applications, including linear systems, optimization, and computational modeling. Common types of matrix equations, such as Sylvester, Lyapunov, and Riccati equations, are widely used in these fields. However, classical approaches are less effective to handle uncertainty in real-world problems. To address this, fuzzy set theory, intuitionistic fuzzy set theory, and neutrosophic set theory offer distinct mathematical frameworks for incorporating uncertainty into matrix equations. This paper provides a comparative review of these three approaches in solving matrix equations with uncertain coefficients. It explores their theoretical foundations, including key definitions, theorems, and arithmetic operations. The strengths and advantages of each theory are highlighted, particularly in terms of handling different degrees of uncertainty. Additionally, an analysis of previous studies on the application of these theories to uncertainty matrix equations is presented, identifying their limitations and areas for improvement. The review emphasizes the potential of neutrosophic set theory, which extends fuzziness and intuitionism, offering a more flexible and comprehensive approach. Finally, recommendations are provided to enhance solution quality by refining existing methodologies and leveraging the strengths of neutrosophic sets for better uncertainty modeling in matrix equations.