Lecture 22: Powers of matrices, introduction to eigenvalues & eigenvectors (Nicholson Section 3.3)
Sub Project: Linear Algebra Lecture Videos
Video Link: Lecture 22: Powers of matrices, introduction to eigenvalues & eigenvectors (Nicholson Section 3.3)
Alternate Video Access via MyMedia | Video Duration: 47:44
Description: Started with a 2x2 matrix and looked at what happened if I applied it over and over again to [1;0]. It’s converging to the vector [3;-2]. What’s with that? If I apply it over and over again to a different vector, I find that the result converges to something. Why? How did I compute A^40 power anyway?
5:25 --- Represented the matrix as a product of three matrices, one of which is diagonal. This made it super-easy to compute A^n and also to figure out where those limiting vectors were coming from.
16:20 --- Introduced the definition of an eigenvector of a linear mapping. Defined eigenvalue and eigenvector-eigenvalue pair.
18:40 --- Did geometric example --- what are the eigenvalues & eigenvectors for reflecting about a line? What do they mean geometrically?
23:10 --- Demonstrated that a nonzero multiple of an eigenvector is also an eigenvector.
27:20 --- Why do we require that eigenvectors be nonzero?
29:15 --- Did geometric example --- what are the eigenvalues & eigenvectors for projecting onto a line? What do they mean geometrically?
33:00 --- What about counterclockwise rotation by theta? Can you find a (real) eigenvector?
34:40 --- If I give you a matrix and a vector, how can you figure out if the vector is an eigenvector? If it is an eigenvector, how can you find its eigenvalue?
37:35 --- Given a matrix, how do I find its eigenvectors and eigenvalues? Tried the natural first idea --- tried to find the eigenvector vector and eigenvalue simultaneously. Got two nonlinear equations in three unknowns. Yikes!
41:15 --- Try to break the problem into two steps. First find the eigenvalues. Subsequently, for each eigenvalue try to find eigenvectors. Explained why we’re looking for lambdas so that det(A-lambda I) = 0.
