Diagonalização de matrizes 3 x 3 e reconhecimento de quádricas

Abstract

This paper deals with the recognition of quadrics using the method of diagonalization of matrices 3 × 3. Earlier it shows the definition of quadrics, the standard equations followed by their names and geometric representations. Then follows the ideas of eigenvalues and eigenvectors of a linear transformation that are the basis for the diagonalization of matrices. Immediately after the linear independence of the eigenvectors is discussed as well as their properties of forming a basis of a vector space. The condition for any square matrix be diagonalizable is shown after, as well as the particularities of a symmetric matrix. The demonstration that all 3 × 3 symmetric matrix is diagonalizable is made from an elegant and elemental matrix approach. Recognition of quadrics is made from basic calculations using some content widely exploited in high school such as matrices, determinants, linear systems and algebraic equations. At the end it presents a way of teaching quadrics in school using educational software Winplot.