Topics in Linear Algebra A Conference at Iowa State University  September 13-14, 2002 Sponsored by the Institute for Mathematics and Its Applications, the International Linear Algebra Society, and Iowa State University

Topics: Matrix Completion Problems - Leslie Hogben , Mathematics, ISU, lhogben@iastate.edu Numerical Ranges - Y. T. Poon , Mathematics, ISU, ytpoon@iastate.edu   Applications of Linear Algebra to Non-Associative Algebra - Irvin Hentzel , Mathematics, ISU, hentzel@iastate.edu   Statistical Applications of Linear Algebra - Huaiqing Wu , Statistics, ISU, isuhwu@iastate.edu

• Matrix Completion Problems-

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 .
 
• Numerical Ranges-

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, systems 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 ideas

9/17/02