2008 IMA PI Summer Program for Graduate Students: Linear Algebra and Applications June 30 - July 25, 2008    Iowa State University   supported by Institute for Mathematics and Its Applications National Science Foundation (DMS-0753009) ISU Department of Mathematics

• Program Overview
• Contents by week, including notes
• Week 1: Linear algebra and applications to combinatorics, taught by Bryan Shader
• Week 2: Numerical Linear Algebra, taught by David S. Watkins
• Week 3: Matrix inequalities in science and engineering, taught by Chi-Kwong Li
• Week 4: Applications of linear algebra to dynamical systems, taught by Fritz Colonius
• Schedule
• Pictures
  
• Photo Album
• Article about the program

Program Description

Contents by each week including notes

Notes of Shader's lectures taken by Olga Pryporova notes1   notes2   notes3   notes4   notes5

Exercises by Bryan Shader:  exercieses1    exercieses2    exercieses3    exercieses4    

Combinatorial matrix theory, encompassing connections between linear algebra, graph theory, and combinatorics, has emerged as a vital area of research over the last few decades, having applications to fields as diverse as biology, chemistry, economics, and computer engineering.

The eigenvalues of a matrix of data play a vital role in many applications. Sometimes the entries of a data matrix are not known exactly. This has led to several areas of qualitative matrix theory, including the study of sign pattern matrices (matrices having entries in {+,- or 0}, used to describe the family of matrices where only the signs of the entries are known). Early work on sign pattern matrices arose from questions in economics and answered the question of what sign patterns require stability, and there has been substantial work on the question of which patterns permit stability, and on sign nonsingularity and sign solvability.

Linear algebra is also an important tool in algebraic combinatorics. For example, spectral graph theory uses the eigenvalues of the adjacency matrix and Laplacian matrix of a graph to provide information about the graph.

Slides by David S. Watkins:   slides1   slides2   slides3   slides4

Notes of Watkins' lectures taken by Olga Pryporova: notes1   notes2   notes3   notes4   notes5

Exercises by David S. Watkins:  exercises

The ability to carry out matrix computations numerically, with accuracy and efficiency, is essential for applications.

This week will survey the most important techniques for solving linear algebra problems numerically, with emphasis on computing eigenvalues and eigenvectors. Methods for solving small to medium-sized problems will be discussed and contrasted with methods for solving large to very large problems. Sensitivity issues and the effects of roundoff and other errors will be discussed.

Topics to be surveyed include: LU decomposition and Gaussian elimination, elementary reflectors, QR decomposition and the Gram-Schmidt process, Schur's theorem, spectral theorem, power method and subspace iteration, Hessenberg matrices, QR algorithm, data structures for handling large matrices, Krylov subspaces and Krylov subspace methods, Arnoldi and Lanczos processes, shift-and-invert strategy, Jacobi-Davidson methods (time permitting), sensitivity and condition numbers, backward stability, effects of roundoff errors, preservation and exploitation of structure.

Lecture notes by Chi-Kwong Li  notes1   notes2   notes3   notes4   notes5

Notes of Li's lectures taken by Olga Pryporova  notes1   notes2   notes3   notes4   notes5

Matrix inequalities have applications to many branches of pure and applied areas, including quantum computing, mathematical biology, perturbation theory, optimal parameters in iterative methods and optimization problems in distance-squared matrices.

Topics discussed in Week 3 will include: Higher rank numerical range and local C-numerical range in quantum dynamics, Perron Frobenius theory and matrix inequalities in population dynamics, Hermitian and skew-Hermitian splitting method in iterative algorithm, selection of optimal parameters for two-by-two block systems and the convergence properties of the Hermitian and skew-Hermitian splitting method, distance matrices in the study of molecular structure, graph layout, and multidimensional scaling (MDS).

Photos