Formas canônicas: considerações e métodos para obtê-las

Abstract

In Linear Algebra, linear operators cannot always be diagonalized, but their matrix representations can be simplified using Canonical Forms, such as the Jordan Canonical Form and the Rational Canonical Form. This study aims to understand how Canonical Forms can simplify the matrix representation of an operator T of a finite-dimensional vector space. We investigate some of the various canonical forms used, highlighting their properties and methods for obtaining them. We also highlight the relevance of these forms in simplifying linear operators, allowing a more detailed analysis of their properties. Based on references such as Hoffman (1970), Herstein (1970), Gonçalves (2006), Coelho (2020), Lima (2006), Lipschutz (2004), among others, we investigate how these tools help in the representation and understanding of operators. Canonical Forms play a central role in identifying the essential characteristics of linear operators, being crucial for both Linear Algebra and Pure Mathematics. Therefore, this study demonstrates the importance of these canonical forms for expanding knowledge about the structure of linear operators, making them indispensable in several areas of Mathematics.