
Linear Algebra and Applications June 30  July 25, 2008 Iowa State University supported by Institute for Mathematics and Its Applications National Science Foundation (DMS0753009) ISU Department of Mathematics 
Fritz Colonius  Institut für Mathematik, University of Augsburg, Germany 
ChiKwong Li 
Mathematics, College of William and Mary 
Bryan L. Shader  Mathematics, University of Wyoming 
David S. Watkins  Mathematics, Washington State
University 
Leslie
Hogben 
Mathematics, Iowa State University 
Wolfgang Kliemann  Mathematics, Iowa State University 
Yiu Tung Poon  Mathematics, Iowa State University 
Jason Grout  Mathematics, Iowa State University 
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.
Week 2: Numerical Linear Algebra, taught by David S. Watkins.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 mediumsized 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 GramSchmidt 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, shiftandinvert strategy, JacobiDavidson methods (time permitting), sensitivity and condition numbers, backward stability, effects of roundoff errors, preservation and exploitation of structure.
Lecture notes by ChiKwong 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 distancesquared matrices.
Topics discussed in Week 3 will include: Higher rank numerical range and local Cnumerical range in quantum dynamics, Perron Frobenius theory and matrix inequalities in population dynamics, Hermitian and skewHermitian splitting method in iterative algorithm, selection of optimal parameters for twobytwo block systems and the convergence properties of the Hermitian and skewHermitian 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.8:00  8:30 
break 
8:30  10:00  lecture by the principal speaker 
10:00  10:30  break 
10:30  noon  lecture by the principal speaker 
noon  1:30  lunch 
1:30  3:30  small group work session or computer tutorial 
3:30  4:00 
break 
4:00  5:00 
Q & A with the principal speaker or guest lecture 