CEU Electronic Theses and Dissertations, 2011
| Author | Pálfia, Miklós |
|---|---|
| Title | On Affine Matrix Means |
| Summary | In this thesis we find all possible matrix means which are midpoint operations on affinely connected manifolds. We characterize certain properties of these manifolds, decide whether they are metrizable or not. It turns out that most of these affinely connected manifolds are not metrizable, so certain geometric procedures used to extend means of two matrices to three or more matrices are only applicable in the case of the geometric mean. All the other symmetric matrix means are not midpoint operations on Riemannian manifolds except the arithmetic and the harmonic mean. However at the same time it turns out that there is a one, real parameter family of affinely connected spaces which have midpoint operations that are in fact symmetric matrix means, and this family is at the same time exhaustive. In the first couple of sections we also cover the theory of operator monotone functions and matrix means. We also investigate detailed properties of the symmetric space corresponding to the geometric mean. |
| Supervisor | Hegedűs, Pál |
| Department | Mathematics MSc |
| Full text | https://www.etd.ceu.edu/2011/palfia_miklos.pdf |
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