Eigenvalues and Eigenvectors: Polynomials and Matrices

This lecture covers the concept of eigenvalues and eigenvectors, focusing on the characteristic polynomial of a matrix. The instructor explains how to determine eigenvalues and eigenvectors, illustrating with examples of 2x2 and 3x3 matrices. The lecture delves into the algebraic and geometric multiplicities of eigenvalues, emphasizing the importance of the characteristic polynomial in determining the number of eigenvalues. The discussion extends to the limitations of using matrix operations like summation and scalar multiplication to calculate eigenvalues. The instructor also highlights the significance of understanding the algebraic and geometric multiplicities of eigenvalues in solving matrix problems.