Lecture
This lecture explores the concept of diagonalizability of matrices by analyzing the relationship between algebraic and geometric multiplicities of eigenvalues. The instructor explains how to determine if a matrix is diagonalizable by comparing these multiplicities and finding linearly independent eigenvectors. Through examples, the lecture demonstrates the process of finding eigenvectors and forming bases for eigenspaces. The importance of understanding the dimensions of eigenspaces and their implications for diagonalizability is emphasized. The lecture concludes by highlighting the key theorem that determines whether a matrix is diagonalizable based on the equality of algebraic and geometric multiplicities of eigenvalues.