## Math 424-524: Advanced Linear Algebra,

Fall 2012

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).

### Exams, Fall, 2012

### Problem sets and exams, Fall 2007

### Older exams: