Diagonalization: Theory and Examples

This lecture delves into the concept of diagonalization of matrices, focusing on eigenvalues and eigenvectors. The instructor explains the conditions for a matrix to be diagonalizable, the importance of linearly independent eigenvectors, and the process of finding the matrix P. Through detailed examples, the lecture demonstrates how to calculate eigenvalues, determine eigenvectors, and construct the diagonal matrix D. The discussion also covers the significance of distinct eigenvalues in diagonalization and provides insights into the relationship between eigenvectors and the kernel of a matrix. The lecture concludes with a corollary highlighting the diagonalizability of matrices with distinct eigenvalues.