What is it about?
This paper presents an alternative approach to diagonalizing 2×2 real matrices without using concepts from vector spaces. Necessary and sufficient conditions for diagonalizability are established, and explicit formulas are given for the diagonal matrix and the corresponding change-of-basis matrix. The method is elementary and accessible to undergraduate students with a background in linear algebra. Applications include the computation of matrix powers and the solution of certain second-order linear recurrence relations.
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Why is it important?
Diagonalization is one of the most important topics in linear algebra, but it is usually introduced using eigenvectors and vector space theory. This work shows that, in the case of 2×2 matrices, diagonalization can be developed by elementary methods. Such an approach may be useful for teaching, allowing students to understand the subject before studying more abstract concepts.
Perspectives
Diagonalization is a fundamental topic in linear algebra, but it is often introduced through abstract concepts such as vector spaces and eigenvectors. In this work, we show that the diagonalization of real 2×2 matrices can be developed using only elementary algebraic techniques. We derive explicit formulas and establish necessary and sufficient conditions for diagonalizability. Besides offering an alternative perspective on the subject, the method provides a useful tool for teaching and has applications to matrix powers and second-order linear recurrence relations.
Josimar da Silva Rocha
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This page is a summary of: Um método não convencional para a diagonalização de matrizes 2x2, Revista Professor de Matemática On line, January 2020, Sociedade Brasileira de Matematica,
DOI: 10.21711/2319023x2020/pmo821.
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