spectral theorem

The spectral theorem states that certain mathematical objects called normal matrices or operators can be broken down into simpler parts using their eigenvalues and eigenvectors. Think of it like decomposing a complex sound into individual notes; similarly, the theorem reveals the fundamental "building blocks" of these matrices. This decomposition allows for easier analysis and understanding of systems, especially in areas like quantum mechanics and signal processing, by representing complex transformations as combinations of simpler, well-understood components.