Eigenvalues and Matrix Equations: A Comprehensive Analysis

This lecture focuses on the determination of positive integers n for which there exist n by n real invertible matrices A and B such that the equation AB - BA = B²A holds. The instructor discusses techniques for solving matrix equations, emphasizing the importance of eigenvalues. The lecture begins with a problem from the IMC 2019, where the instructor guides the audience through the process of rewriting the equation to identify common factors. The discussion highlights the relationship between eigenvalues of matrices and their implications on the structure of the matrices involved. The instructor illustrates how to construct examples for even values of n and explores the implications of eigenvalues being real and distinct. The lecture culminates in a deeper understanding of the Cayley-Hamilton theorem and its application in proving the existence of matrices with specific eigenvalue properties. The session concludes with a discussion on the implications of these findings in the context of linear algebra and matrix theory.