Workshop: Eigenvalues and Eigenvectors

Based on the success of last week's workshop on determinants, we're going to take the same approach with the next big topic in the course: eigenvalues and eigenvectors. As before, there are four basic components to the workshop: the reading/viewing assignments to be done outside of class, the guided practice exercises also to be done outside of class, the quiz we will have in class, and then the in-class activities which you will start in class and be allowed to finish later. This workshop will take place on Friday, April 15.


We're going to return to the concept of linear transformations for a bit, which we can think of as actions or movements that we perform on R^2 or R^3 (or larger spaces). For example, a rotation about the origin of 45 degrees is a linear transformation that maps R^2 to itself. So is a flip of R^2 around the line y = x. But these two transformations are different in one important way: The rotation mapping leaves no vectors fixed in place (except the zero vector), whereas the flip mapping does fix some vectors in place (namely, all vectors on either the lines y = x or y = -x -- try this for yourself to see). So we're going to ask the question: Given a linear transformation, which vectors are merely fixed or rescaled in place? This will lead to the central concepts of eigenvalues and eigenvectors, which are essential elements of applied mathematics in fields as diverse as economics, engineering, and computing.


You have a little viewing and little reading to do this time. First of all, watch these five screencasts (homemade this time, not from Khan Academy):

Once you've viewed these and feel satisfied that you've understood the ideas (you can gauge that understanding by working on the Guided Practice exercises), go to your textbook and:
  • Read pages 302--305 in section 5.1, just skimming to get a review or second look at the concepts addressed in the screencasts, then
  • Read pages 306--307 for understanding, particularly the two theorems that appear on those pages. Read both the theorems and their proofs and make sure you understand each.


By class time on Friday, each student should be able to perform the following tasks and respond correctly on a 5-minute quiz:
  • Find the standard matrix for a linear transformation.
  • State the definition of "eigenvalue", "eigenvector", and "eigenspace".
  • Given a matrix A, tell if a vector v is an eigenvector of A.
  • Given a matrix A, tell if a scalar λ is an eigenvalue of A.
  • Given an eigenvalue λ of a matrix, find a basis for the eigenspace associated with that eigenvalue.
  • Find an eigenvector corresponding to a given eigenvalue.
  • Find (quickly!) the eigenvalues of a triangular matrix.


Exercise 1: Give the standard matrix for the transformation from R^2 to R^2 that flips vectors across the line y = x and then flips them across the x-axis. This should be ONE matrix that performs both actions.

Exercise 2: Consider the matrix

  • Verify that the vector v = [2; 1] is an eigenvector of A by performing a multiplication step.
  • Verify that the vector v = [2; 2] is NOT an eigenvector of A, by performing a multiplication step.
  • Verify that λ = 4 is an eigenvalue of A. (Hint: Look at the work in the first bullet point.)

Exercise 3: Consider the matrix

One of the eigenvalues of this matrix is λ = -1. Find one eigenvector corresponding to this eigenvalue and check your work using matrix multiplication. (Hint: If -1 is an eigenvalue, then there is a nonzero vector x such that Bx = -x. That is, there is a nonzero solution to the equation (B + I)x = 0 where I is the 2x2 identity matrix. Find one such solution using basic linear algebra techniques.)


Please hand in a typed-up or neatly handwritten submission that contains all your work on these tasks by class time on Friday, April 15. Since our week is somewhat compressed, you do not need to hand any of these exercises in earlier this week. But do not wait until Thursday night to start watching the videos! Start those early and work a little on them every day. Please do all work by hand this time, with no MATLAB. You can check your work with MATLAB if you wish.


Lastly: You may ask me (Prof. Talbert) questions on this material AT ANY TIME and THROUGH ANY MEDIUM you wish. I am a resource to help you navigate through the video material and help you translate this into understanding. Please use me!