Module 6: Eigenvalues and Eigenvectors
Discover the special vectors that a matrix merely scales, learn to compute eigenvalues and eigenvectors, and apply diagonalization to solve real-world problems.
Module Lessons
1
Eigenvalues and Eigenvectors: Introduction
Define eigenvalues and eigenvectors, understand the geometric meaning of Av = lambda v, and compute them for 2x2 matrices.
2
The Characteristic Equation
Derive and solve det(A - lambda I) = 0 for 2x2 and 3x3 matrices, and distinguish algebraic from geometric multiplicity.
3
Eigenspaces and Diagonalization
Compute eigenspaces, determine when a matrix is diagonalizable, and carry out the P^{-1}AP = D procedure.
4
Applications: Markov Chains and Matrix Powers
Use diagonalization to compute A^n efficiently, find steady-state vectors for Markov chains, and model dynamical systems.