| Me: | Mark Steinberger |
| Office: | ES136A |
| Hours: | MWF 12:35-1:30 |
| Email: | mark@albany.edu. Please put Math 424-524 in the subject line. |
| Home page: | http://math.albany.edu/~mark |
| Text: | Algebra, by Mark Steinberger, (the latest version) available online in pdf format |
| Review material: | web site for my Elementary Linear Algebra course |
The purpose of this class is to develop tools for determining whether two n by n matrices are similar: we say that A and B are similar if there is an invertible matrix P with
Basically, two matrices are similar if they represent the same linear transformation looked at from two different points of view. (We'll discuss these concepts in depth, of course.)
The techniques we will use are the Rational Canonical Form and the Jordan Canonical Form, which are special types of matrices. We will see that each matrix is similar to a unique canonical form. Thus two matrices are similar if and only if they have the same canonical form. We will also see how to find the Jordan Canonical Form for a particular matrix, providing an algorithm for solving the problem. (The one nonalgorithmic step is factoring the characteristic polynomial. Recall that there is no formula for factoring polynomials of degree greater than 4. We will discuss this issue in class.)
An additional benefit of the Jordan Canonical Form is that we can use it to uncover the geometric effect of the underlying linear transformation. This generalizes the study of diagonalizable matrices given in many Elementary Linear Algebra courses.
The heart of the course is Chapter 10 of Algebra, in which the canonical forms are studied. To get there, we will need material from Chapters 7 and 8, and some review material from the Math 220 web site.
All exams are take-home. The final is due on the last day of class. Some of the graded material will be designated as problem sets.
Homework will be given and discussed in class, and is an important component of learning. Please do all the homework problems. If there is anything you are not understanding and for which there is not time to discuss in sufficient detail in class, please be sure to come to office hours. Additional hours are possible, particularly later in the day on MWF. Please ask questions freely and make liberal use of office hours. We often learn best by talking rather than simply studying on our own.