Abordagem teórica de autovalores e autovetores: estudo de caso de quádricas na segunda e terceira dimensão

Abstract

This work deals with algebraic concepts relating to eigenvalues and eigenvectors of a transformation T : V →V, V ⊂ R. First, was introduced the subject stating the reason for researching and applying eigenvalues and eigenvectors of the quadric, and bring the motives and objectives of the work. Then, in the prologue section, key concepts were reminded about vector spaces and linear transformation, then to define the main object of our study. In the Eigenvalues and Eigenvectors section bring the theoretical basis on invariant subspaces, autoespace of T : V →V besides the relevant relations about eigenvalues and eigenvectors, and diagonalization transformation matrix T. After this section, was raised a whole context to algebraic demonstration of the Spectral Theorem and present the bilinear forms, in particular the bilinear quadratic form. After this section, was presented the theoretical application of the eigenvalues and eigenvectors in quadrics in R2 and R3 and shown as important the eigenvalues and eigenvectors are to analysis the algebrics classification of quadrics. Finally, following the last section which brings the conclusion of this work.