Teorema Decomposição Espectral e suas aplicações

Abstract

This work presents a structured study on the diagonalization of linear operators in finite-dimensional vector spaces. Initially, fundamental concepts of vector spaces, bases, dimension, and linear transformations are reviewed, with an emphasis on the Rank-Nullity Theorem (Teorema do Núcleo e da Imagem). The investigation then delves into the geometry of spaces equipped with an inner product, both real (Euclidean) and complex (Unitary), establishing the definitions of orthogonality, adjunction, and normal operators. The core focus of this monograph is the Spectral Theorem for Self-Adjoint Operators. It is rigorously demonstrated that such operators possess purely real eigenvalues and that eigenvectors corresponding to distinct eigenvalues are orthogonal, guaranteeing the existence of an orthonormal basis that diagonalizes the operator. The theory is illustrated through practical examples, including the analysis of symmetric matrices and differential operators in polynomial spaces, highlighting the elegance and applicability of spectral decomposition.