Matrix completion problems are an important part of the
growing field of combinatorial matrix theory. A partial matrix is a
matrix in which some entries are specified and others are not. A completion
of a partial matrix is a specific choice of values for the unspecified entries.
A matrix completion problem asks whether a partial matrix (that meets certain
necessary conditions) has a completion with a desired property, or whether
all partial matrices specifying a certain pattern of entries have completion
of a desired type. Applications arise in situations where some data
are known but other data are either not collected or lost, and it is known
that the full data matrix must have a certain property. Such situations
include seismic reconstruction problems and data transmission and coding,
etc. There is an active
matrix completions research group
at Iowa State University that maintains a
matrix completions web site
The Toeplitz-Hausdorff Theorem on the convexity of numerical
range has been generalized in many directions. These results on
generalized numerical ranges turn out to be related to many other areas:
1. The existence of a positive definite matrix in a given
subspace of hermitian matrices.
2. The existence of a matrix for which the eigenvalues
lie in a specific set.
3. Representation of completely positive maps.
The numerical range also has a lot of applications to other
subjects such as quantum physics, numerical analysis, perturbation theory,
Applications of Linear Algebra to Non-Associative Algebra-
Commercial algebra packages like Mathematica, MatLab, and
Maple are not intrinsically designed for problems in nonassociative algebra
since they are designed to work with real numbers, which are associative and
commutative. These packages, however, can be used very productively
in nonassociative algebra studies by taking advantage of their linear algebra
routines as well as other built in functions. In this conference we
invite the participants to describe their techniques for using standard computer
packages on questions in nonassociative algebra. We are looking for
insightful ways which employ the standard given routines on problems where
the interface between the problem and the routine is accomplished by ingenuity.
The algebra package can be something like ALBERT, which is designed for nonassociative
algebra, or it can be any other commercially available package.
Statistical Applications of Linear Algebra-
Linear algebra is essential for development of statistical
theory and methods, especially in such important statistical areas as linear
models, multivariate analysis, and experimental designs. There is an increasing
interest in research in linear algebra with special emphasis on statistical
applications. This conference will provide opportunities for statisticians
and algebraists to discuss the newest developments in the field and to exchange