Diagonalização das formas quadráticas e classificação das cônicas e quádricas

Abstract

Conics and quadrics can be classified using eigenvalues and eigenvectors of their respective associated matrices. For conics, the matrix is obtained from the general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. For three-dimensional quadrics, the matrix is derived from the general equation $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$. The classification depends on the eigenvalues and eigenvectors of the matrix. For conics, if all eigenvalues are of the same sign, the conic is classified as an ellipse or a point. If an eigenvalue is zero, it is a parabola. If the eigenvalues have opposite signs, we have a hyperbola. For quadrics, the nature of the surface (ellipsoid, hyperboloid, paraboloid, etc.) is determined by the eigenvalues and eigenvectors of the matrix. The exact interpretation may vary depending on the dimension (2D or 3D) and the specific characteristics of the equations.