Operadores lineares auto-adjuntos e o teorema espectral.

Abstract

In this work, we present one of the most relevant results of Linear Algebra, the Spectral Theorem. This Theorem tells us that there exists a basis of V , where V is a real vector space of nite dimension such that certain linear operators T : V → V have that matrix with respect to this base of the simplest possible form, the dia- gonal matrix. There are cases of unit operators, normal operators and self-adjoint compact operators in Hilbert spaces, which have this property, but we will empha- size only the self-adjoint operators. When it's possible to determine this basis, we say that the linear operator T is diagonalizable. Thus, the Spectral theorem tells us that every self-adjoint linear operator is diagonalizable, that is, the matrix that represents it is a diagonal matrix and still more, the elements of the main diagonal are the eigenvalues associated with the operator. The main objective of this work is to present and demonstrate the Spectral Theorem and to bring examples and ap- plications of it. For this, we will cover the following subjects: self-adjoint operators, eigenvalues and eigenvectors and diagonalizable operators.