Masterproef T808 : Computing the matrix geometric mean with preservation of matrix properties

Informatie:	Raf Vandebril
Promotoren:	Raf Vandebril
Begeleider:	Bart Vandereicken

Numerieke Approximatie en Lineaire Algebra Groep

The geometric mean of some positive numbers is defined as the n‐th root of the product of these numbers. It is clear that this mean does not depend on the order of the numbers itself. Unfortunately due to the noncommutativity of matrices, it is not straightforward to generalize the geometric mean to positive definite matrices while satisfying this property. Several techniques exist, however, to compute the matrix geometric mean, thereby exploiting the fact that positive definite matrices lie on a manifold. There are iterative techniques on manifolds (e.g. constructing triangles in triangles where the three matrices denote the corners until we converge to a center of the triangle) but also techniques on computing midpoints of geodesics on the manifold and their intersections. Appealing is the fact that for a limited number of matrices a visual representation of many of the algorithms is possible.

An extra difficulty arises when one wants to preserve properties of matrices. Consider for example an application that can be described by Toeplitz matrices. When computing the mean one wants the mean to be Toeplitz as well. Otherwise the corresponding result has no physical meaning anymore.

The students has to study the problem of computing the matrix geometric mean. He has to identify the problems, and see how this problem can be formulated over a manifold. The student has to study the various algorithms, implement them and compare their performance and accuracy, analyze their stability, drawbacks advantages and so forth. Finally the student has to investigate which matrix properties can be translated to submanifolds and see for which matrix properties it is straightforward to retain the matrix property when computing the matrix geometric mean.

The student needs to be familiar with numerical linear algebra and needs to be able to develop software in e.g. Matlab, C++ or Fortran.