Linear Algebra and Applications

This program is primarily for graduate students of IMA Participating Institutions. The NSF may provide support for a limited number of students at other US universities. In order to participate, students need to fill out the application form and provide a letter of nomination.

From Monday, June 30 through Friday, July 25, 2008, Iowa State University will be the host of the Institute for Mathematics and its Applications (IMA) Summer Graduate Program in Mathematics. The course will concentrate on Linear Algebra and Applications.

Program Description:

Linear algebra is a subject of central importance in both mathematics and a variety of other disciplines. Research in linear algebra is very active, with a wealth of applications, and linear algebra is also a powerful tool for research in other areas. The program will run for four weeks and cover linear algebra, numerical linear algebra, and applications. Four topics will be covered, one per week. Each of the speakers will present open research questions toward the end of the week, and it is our goal to have each student involved in at least one research problem during the course of the workshop.

• Week 1: Linear algebra and applications to combinatorics, taught by Bryan Shader, University of Wyoming

  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

• Week 2: Numerical linear algebra, taught by David S. Watkins, Washington State University

  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

• Week 3: Matrix inequalities in science and engineering, taught by Chi-Kwong Li, College of William and Mary

  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)

• Week 4: Applications of linear algebra to dynamical systems, taught by Fritz Colonius, University of Augsburg

  Linear algebra is a key tool in the study of ordinary differential equations, including the explicit form of solutions to linear equations, linearization theory, and results on invariant manifolds and the Grobman-Hartman theorem.

  The connection actually goes much deeper, as classes of matrices can be characterized by concepts from dynamical systems, such as Ck conjugacies and equivalences of flows in Rn associated with linear ODEs. Probing this connection further, for a linear ODE one can analyze its radial component (eigenvalues, Floquet exponents, Lyapunov exponents) and its angular component on the sphere, leading to an interesting introduction to attractor-repeller pairs and Morse decompositions. These topics will motivate the contents of Week 4.

  With this background it is now possible to use ideas from linear algebra for a variety of dynamic problems in the sciences and engineering. We will concentrate on control theory, specifically on questions of robust stability and stabilizability in engineering systems, including (linear and nonlinear) stability radii, characterization of stabilizability for uncertain systems, and - if time permits - on the global behavior of randomly perturbed systems.

  Engineering systems to be considered include tank reactors, electric power systems, and nonlinear oscillators.

It is expected that participants will have had standard undergraduate courses in linear algebra, numerical analysis and ordinary differential equations. Some exposure to canonical forms and matrix factorization would be helpful, although these topics will be reviewed as needed, especially in week one.