Diagonalização de operadores e formas lineares: cônicas e quádricas

Abstract

In this work we will make applications of Linear Algebra to conics and quadrics. We will discuss the rotation and translation of these particular curves and surfaces and understand why the rotation process relates to with matrix diagonalization. We will see that the quadratic form of a conic or a quadric is associated to a symmetric bilinear shape and therefore to a self-adjoint linear operator. Given this fact and through theorems of Linear Algebra, we will show that the vector spaces R2 and R3 admit an orthonormal basis consisting of characteristics vectors. That basis that will be used to create a new system of orthogonal axis under which the conical or quadric will no longer be rotated, that is, the matrix of the quadratic form will become diagonal and with that property the quadratic form will lose mixed terms. In addition, we will bring examples of conics and quadrics in their canonical forms and we will also show the degenerate cases. We will still show some examples of conics and an example of quadric when they are rotated and translated in relation to the canonical cartesian system.