Lecture 18 Eigenvalues and eigenvectors

0:01 direction of eigenvector of matrix A is fixed under map T(v)=Av
3:05 (1,1)^T is an eigenvector of example matrix A (graphical)
5:56 (1,-1)^T is an eigenvector of example matrix A (graphical)
7:17 definition of eigenvalue and eigenvector
9:12 verification that (1,1)^, (1,-1)^T are eigenvectors of example matrix A (using definition)
11:45 eigenvalues determined from characteristic polynomial
13:11 characteristic polynomial of example 2 x 2 matrix
16:05 eigenvalues obtained from characteristic equation
17:18 example of obtaining eigenvector for a given eigenvalue \lambda=1
17:40 rearrange Av=(-1)v as 0=((-1)I_2-A)v
22:20 eigenvector is a nonzero solution of the matrix equation 0=((-1)I_2-A)v
22:45 review of null space of a matrix
24:15 zero column of augmented matrix of homogeneous system is redundant
27:14 an eigenvector is a nonzero vector in the null space of a certain matrix
29:03 obtain eigenvector (1,-1)^T of example matrix A
31:01 obtain eigenvector (1,1)^T of example matrix A
32:14 obtain eigenvalues of example 3 x 3 matrix
35:44 obtain eigenvector corresponding to eigenvalue -1 of example 3 x 3 matrix
39:21 obtain eigenvector corresponding to eigenvalue 1 of example 3 x 3 matrix