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ELA

CONSTRUCTING MATRIX GEOMETRIC MEANS

∗

FEDERICO POLONI†

Abstract. In this paper, we analyze the process of “assembling” new matrix geometric means from existing ones, through function composition or limit processes. We show that forn= 4 a new matrix mean exists which is simpler to compute than the existing ones. Moreover, we show that forn >4 the existing proving strategies cannot provide a mean computationally simpler than the

existing ones.

Key words. Matrix geometric mean, Positive definite matrix, Invariance properties, Groups of permutations.

AMS subject classifications.65F30, 15A48, 47A64, 20B35.

∗Received by the editors October 12, 2009. Accepted for publication July 13, 2010. Handling

Editor: Ravindra B. Bapat.

†Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy ([email protected]). Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society Volume 20, pp. 419-435, July 2010

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