# Matrix Analysis

Math 623
Spring 2013
Course number: 21810
Meeting Tu Th 9:30 - 10:45
GMCS 328
San Diego State University
Last Class: Tu. May 7.
Final Exam: Fri. Th. May 16, 8:00-10:00.

Professor: Mike O'Sullivan
Web page: http://www.rohan.sdsu.edu/~mosulliv/Copurses/matrix13s.html
Email: mosullivan@mail.sdsu.edu
Office: GMCS #579, ext. 594-6697
Office Hours: Tu Th 1-3:00.
I will usually be available Tu Th 11-12. On other days I am often available in the afternoon, but it is best to email me in advance to schedule a meeting.

## References

Horn, Johnson Matrix Analysis Cambridge University Press, 1st Ed. (1985) or 2nd Ed. (2012).

This is a standard and highly regarded reference on the subject. While we will not follow it closely, nearly everything we study will be found, in great depth, in this book.

It is worth having an undergraduate text handy, since Horn's book is quite intensive. Here are references that I recommend, but anything you used in a previous course should be sufficient.

Strang, Linear Algebra and Its Applications Any edition. A widely used, very readable book. I used it to prepare notes on orthogonal subspaces and the Gram-Schmidt process. Strang's lectures are also available at MIT's website.

Axler, Linear Algebra Done Right Springer. I used this for a reference on Jordan canonical form--developing it without the use of determinants.

Beezer, A First Course in Linear Algebra. This is freely available. The web version has embedded Sage code so that you can do computational experiments within the text of the book.

## Prerequisites

A good understanding of linear algebra as in Math 524. I will assume you've seen the following material. We will cover it at relatively quickly, and at greater depth.
• Solution of linear systems. Row space, nullspace of a matrix
• Vector spaces, linear independence and bases, subspaces.
• Eigenvalues, eigenspaces, diagonalization, and change of basis.
• Polynomial Ring in One Variable: Divisibility and unique factorization, the correspondence between factors and roots.

## Course Description

The are numerous interesting topics that we can study in this course. I expect to introduce the most fundamental results from the following areas,in roughly the order listed.
1. Solution of linear systems and matrix algebra.
2. Vector spaces and subspaces.
3. Orthogonality and Gram Schmidt.
4. Unitary similarity.
5. Eigenvectors, eigenvalues, diagonalization and Jordan canonical form.
6. Hermitian, and symmetric matrices.
7. Location of eigenvalues.
8. Positive definite matrices.
9. Nonnegative matrices and the Perron-Frobenius theorem.
If time allows we can explore some topics in more depth. I'm quite willing to follow the interests of the class.