This paper explores the extension of matrix eigenvalue theory to the spectral theory of operators on Banach and Hilbert spaces, focusing on finite-dimensional Hilbert spaces as a foundational step. Since matrices uniquely represent linear operators in finite-dimensional spaces, the relationship between linear operators and matrices is used to bridge the understanding of spectral theory. The study begins with a detailed examination of the spectrum of an operator, highlighting its properties and implications. The core contribution of this work lies in addressing a challenging problem: deriving the decision-maker's weight vector from concepts rooted in spectral theory. This approach is applied to solve Multiple Attribute Group Decision Making (MAGDM) problems, demonstrating its theoretical robustness and practical relevance. Furthermore, the same MAGDM problem is tackled using well-known methods from Artificial Neural Networks (ANNs). A comparative analysis is conducted to evaluate the practical aspects, productivity, and advantages of the proposed methods compared with existing solutions. This comprehensive investigation aims to provide deeper insights into the interplay among spectral theory, decision-making, and ANN techniques, advancing both mathematical theory and practical applications.