## Schedule

Tuesday and Thursday, 11:30-12:45, 202 Saint John's College.## Course program and material

### Bibliographic resources

We shall be using a combination of lecture notes and extracts from books. Most books used are available to you electronically through the libraries (link). From within campus, following the links posted below will take you directly to a page from which you can access the book content. From outside campus, please see this page (link) for information on how to set up the Proxy Bookmarklet. Note that for many of the books published by Springer, you can purchase a print copy of the book for USD 24.95 (shipping included).### Course content

**6/9.**"Matrices are everywhere": examples of theorems and settings in which matrices appear, including linear systems of difference equations, linear systems of ODE, nonlinear ODE and the stable manifold theorem, Markov chains, graph theory..**11/9.**Review of basic concepts in matrices, notation. Based on Zhang,*Matrix Theory*(link), chapter 1 and sections 2.1, 2.2 and 2.3.**13/9.**Review of basic concepts (continued).**18/9.**Gershgorin's circle theorem. See Section 2.6 in Zhang or Chapter 1 in the book by Varga,*Gersgorin and his circles*(link).**20/9.**Gershgorin's circle theorem (continued).**25/9.**Gershgorin's circle theorem (continued). Some matlab code: circle_for_plot.m, Gershgorin_disks.m and plot_Gerhgorin_disks.m.**9/10.**Applications of the Jordan canonical form in ODE (see slides shown in class here).**10/10.**Proof of the Jordan canonical form theorem. Several different methods of proof can be found in the secure area.**11/10.**Markov chains (see slides shown in class here).**16-18/10**. The Perron-Frobenius theorem. The version proved in class is from Seneta,*Non-negative Matrices and Markov Chains*-link-, Chapter 1.**23/10.**A synthetic view of the Perron-Frobenius theorem (from Keyfitz and Caswell,*Applied Mathematical Demography*-link-, section 7.2.1.4). Some results about irreducibility and primitivity. Used Horn and Johnson,*Matrix Analysis*(see, e.g., here).**30/10.**Weak and strong ergodic theorems. See Golubitsky, Keeler and Rothschild, Theoretical Population Biology, 1975 (link).**13/11.**Adjacency matrices (of graphs) and some properties. See Bapat,*Graphs and Matrices*(link), Chapter 3.**20/11.**Eigenvalues interlacing (Bapat, Chapter 1) and results on the Chromatic number (Bapat, Chapter 3).

### Computer programming

- Introduction to computer software: octave (a GPL -free- MatLab equivalent), MatLab (file) and maxima (a GPL -free- Maple equivalent).

## Tests and Assignments

### Assignments

- Assignment 1. Due Friday, September 28 at 16:30 in the Department of Mathematics office or electronically before midnight. (link)
- Assignment 2. Due Monday, November 5 before midnight -electronically only, written assignments will not be accepted. (link)
- Assignment 3. Due Monday, December 3 before 16:30 in the Department of Mathematics office or electronically before midnight. Please send code electronically in any case. (link)