Applications of the Spectral Theorem: Utilizing Eigenvalues and Eigenvectors

Abstract

This paper explores the applications and principles of both the real and complex spectral theorems, cornerstones of linear algebra and functional analysis. We begin with an overview of the spectral theorem and delve into the pivotal roles of eigenvalues and eigenvectors, culminating in a detailed proof of the theorem. A discerning comparison is made between the real and complex versions; notably, the latter seamlessly integrates with the fundamental theorem of algebra, an attribute absent in the former. The ensuing sections illuminate practical applications. The real spectral theorem emerges as instrumental in multivariable calculus, notably in the second derivative test, in data-driven techniques such as Principal Component Analysis (PCA), and in electrical network analyses via Kirchhoff’s matrix. The complex variant takes center stage in quantum mechanics, illuminating the Schrödinger Equation. This paper underscores the spectral theorem’s profound relevance in diverse theoretical and practical arenas.