Spectral decomposition

Spectral decomposition is a mathematical process that breaks down a matrix or linear operator into simpler components based on its eigenvalues and eigenvectors. Think of it as identifying the fundamental "building blocks" that make up a complex system. This method allows us to analyze and understand the behavior of systems like vibrations, signals, or transformations by expressing them as a sum of simpler, clearer parts. In essence, spectral decomposition reveals the core features of a system in a way that makes its analysis and computation more straightforward.