This section finishes the story of eigenvalues and eigenvectors when the characteristic polynomial has real roots. The drill is the same as in 4.1 EXCEPT for eigenvalues that are multiple roots of the polynomial. In this case, instead of finding one non-zero eigenvector, the eigenspace may have dimension > 1 and so you will find a basis of the eigenspace. We did examples in class on Wed 11/26, but we did not introduce the terminology. If an eigenvalue c is a root with multiplicity k, we have two numbers, the algebraic multiplicity (this comes from the polynomial) and the geometric multiplicity (the dimension of the eigenspace). Read the definitions on page 308 CAREFULLY. Also read the definition of DEFECTIVE MATRIX.
DO the problems assigned
below.
The key idea of the topic of difference equations is that we can raise a matrix to a high power by writing vectors as linear combinations of a basis of eigenvectors. This extends some work we did before in the Linear Transformation chapter. Markov chains are merely a special class of matrices with (thankfully) a very predictable eigenvalue = 1. They come up whenever we are dealing with 100% of something that gets rearranged.
We may do an example of a DE in class, but you are NOT responsible for DEs on the Final Exam.
DO: the problems assigned below.
This section is really about writing a formula for a linear transformation in non-standard coordinates. The key for diagonalization is a matrix whose columns are a basis of eigenvectors. We will focus on the 2x2 case for problems.
DO: the problems assigned below,
plus maybe try out a couple of others.
This section is really optional, but it does provide an easier way to compute determinants using row operations that some of you may want to check out. If you are asked to compute a determinant on the final, you can choose your method.
DO: Examples, pp. 293-296
Note: Many of these are ODD-numbered problems so the answer is in the back of the book. That is good because you can check your answer. But of course you need to show the work you did to get the answer, since the answer itself is already given.