Linear Algebra and Applications, Iowa
State University, June 30July 25, 2008
Contents for each week
 Week 1:
Linear algebra and applications to combinatorics,
taught by
Bryan Shader.
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.
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.
 Week 3:
Matrix inequalities in science and engineering, taught
by ChiKwong Li.
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.
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 GrobmanHartman theorem.
The connection actually goes
much
deeper, as classes of matrices can be characterized by concepts
from
dynamical systems, such as C^{k} conjugacies and equivalences of
flows
in R^{n} 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 attractorrepeller
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.
 Review material in week 1:
Topics from graduate linear algebra and basic
graph theory
that will be needed will be reviewed without proof but with
examples and
illustrations of use.
This includes Jordan and real Jordan
canonical form,
spectral theory of normal matrices, properties of unitary and
Hermitian
matrices, nonnegative and stochastic matrices, basic
terminology of
graphs and digraphs and connections with matrices.

