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## Math 424-524: Advanced Linear Algebra, Fall 2012

 Office: ES 136A Hours: MWF 1:40-2:35 and by arrangement Email: mark@albany.edu. Please put Math 424-524 in the subject line. Text: Algebra, Mark Steinberger, current edition online. Review material: web site for my Elementary Linear Algebra course. A course in low-dimensional geometry, Mark Steinberger, available online.

### Syllabus

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

PAP-1=B.

Basically, two matrices are similar if they represent the same linear transformation looked at from two different points of view. (In particular, the transformations differ by a linear change of variables. 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 rational canonical form. Thus two matrices are similar if and only if they have the same rational canonical form. We will also see how to find the rational 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.)

The Jordan canonical form exists if and only if the characteristic polynomial has no irreducible factors of degree greater than 1. In this case, it has properties equivalent to those of the rational canonical form, and, in addition, gives insight into the geometric effect of the underlying linear transformation. Jordan canonical form 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.

This course is heavy on problem solving. There will be challenging problems in the problem sets, and examples will be posted and/or discussed in class. There will be plenty of specific problem-solving challenges.

We will spend significant time talking about theory, because the theory is essential in developing problem solving skills.

You are strongly encouraged to discuss this material with each other and with me, both in office hours and in class. Verbalizing mathematical questions is a very useful step toward understanding them. Classroom discussion is strongly encouraged. Please ask questions! If there is something you don't understand or can't follow, there will be a number of other people in the class in the same boat. So a number of people will benefit if you ask.

It is very important to stay current with the material. If you fall behind, it will be hard to catch up. And if you are having trouble, please do come to office hours early on. If you leave it until the last minute, you probably won't be able to learn it in time.

But office hours are not only for those who have fallen behind. Office hours are extremely helpful for learning and I seriously enjoy discussing the material with students and helping them learn. It is especially useful to work with a group of students. The synergy really helps everyone learn. If there is a small group, we will work in my office, ES136C. With larger groups we will work in ES135 (close by).