46. Understanding Linear Algebra 4.2: Introduction of the Characteristic Polynomial and Eigenvalues

In this video, we begin learning how to find eigenvalues of a matrix. Starting from the definition of an eigenvector and eigenvalue, we derive one of the most important equations in linear algebra: det(A - λI) = 0 This equation, known as the characteristic equation, allows us to determine the eigenvalues of a matrix by finding the roots of its characteristic polynomial. Along the way, we connect eigenvalues to homogeneous systems, invertibility, and determinants, showing why the matrix A - λI must be singular whenever λ is an eigenvalue.