Algebra III 69.317*C

Fall 2001

Tel: 520-2600 x 2165

Classes Tuesday 830-1000 Friday 1000-1130 SA518

Those wishing to have their own copy of MATLAB can purchase STUDENT MATLAB at the bookstore.Ê It can also be purchased for $99 US from MATHWORKS:
http://www.mathworks.com/products/studentversion/buy_now.shtml

Students with Disabilities:
Students with disabilities requiring academic accommodations are encouraged to contact a coordinator at the Paul Menton Centre to complete the necessary letters of accommodation. Then, make an appointment to come and discuss your needs with me at least two weeks prior to the first test. This is to ensure sufficient time to make the necessary accommodation arrangement.

Chapter 0: Review (brief). Students are expected to know the material from 69.112 or 69.217 including: determinants, vector spaces, linear independence, basis, dimension, rank, and invertibility. We will frequently discuss vector spaces and matrices over the Field of Complex Numbers. You should review your previous courses during the first 2 weeks of this course.

Chapter 1. Eigenvalues, eigenvectors and similarity. This material will be of significant importance throughout the course. Students should know how to find eigenvalues of 2x2 and 3x3 matrices by hand, and will quickly learn how to do this using MATLAB. We will discuss characteristic polynomials, diagonalizability and similarity.

Chapter 2. Unitary equivalence and normal matrices. The main result is Schurâs unitary triangularization theorem that says that given an nxn matrix A you can find unitary (orthogonal if A is real) matrices U such that U*AU=T, where T is upper triangular and whose diagonal elements are the eigenvalues of A. (Note U* means take the complex conjugate of U and transpose. We call U* the Hermitian adjoint of U). This is intuitively not hard; you just use the Gram Schmidt process from 69117* repeatedly. From this it easily follows that we can diagonalize a real symmetric, or in general, any Hermitian matrix A ( i.e., A=A*). The Cayley Hamilton theorem is an easy consequence; i.e., if the characteristic polynomial of A is c(x)=0, then the matrix polynomial that you get when you substitute the matrix A for x, c(A)=0 too. The QR factorization result follows; this says that the process from your first linear algebra course of reducing a matrix to reduced row echelon form can be done with unitary (or orthogonal) matrices. This is more stable and accurate when programmed onto a digital computer than the methods learned earlier.
The idea here is to revisit the facts that you learned in your first Linear Algebra course and put them in a more general framework so that we can build on the information that you already know. We will do a lot of computation here using MATLAB. The purpose is to show you better tools for attacking two fundamental problems: accurately solving linear systems of equations and finding eigenvalues of a matrix.

Chapter 3. Canonical forms. We will look at the Jordan canonical form to learn when a matrix really is diagonalizable. Recall that in your previous course you were probably told that weâll look at this problem in a later course. This is that course.

Chapter 4. Hermitian and symmetric matrices. These correspond to the set of matrices with real eigenvalues. These arise very naturally in important problems of Physics and Engineering and are of special interest to us. We will study the lovely spectral theorem and learn how the eigenvalue spectrum of the sum of 2 matrices A and B depends on the eigenvalues of A and B themselves (note: this is a subject of active research, and there have been recent important developments concerning the Horn conjectures, named after one of the coauthors of the text.)
As time permits, we will study selected topics from the following chapters:

Chapter 5. Norms for vectors and matrices. We describe how to use simple analysis to put measures on vector spaces, e.g., what is the distance between 2 matrices? We relate some norms of a matrix to its eigenvalues.

Chapter 6. Location and perturbation of eigenvalues. The wonderful result of Gersgorin is that the eigenvalues of a matrix are located within the union of a set of discs on the complex plane surrounding the diagonal elements of the matrix A, and whose radii are given by the sums of the absolute values of the other elements in the rows of the matrix (i.e., excluding the diagonal elements).

Chapter 7. Positive definite matrices. If we just look at the set of matrices with positive real eigenvalues, then there are many fascinating results just waiting for us. We encounter several new important matrix decompositions, such as the singular value decomposition. These are crucial for accurate computer algorithms for solving large systems of equations.