Means of positive matrices: Geometry and a conjecture

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Abstract

Means of positive numbers are well-know but the theory of matrix means due to Kubo and Ando is less known. The lecture gives a short introduction to means, the emphasis is on matrices. It is shown that any two-variable-mean of matrices can be extended to more variables. The n-variable-mean Mn(A1, A2, …, An) is defined by a symmetrization procedure when the n-tuple (A1, A2, …, An) is ordered, it is continuous and monotone in each variable. The geometric mean of matrices has a nice interpretation in terms of an information geometry and the ordering of the n-tuple is not necessary for the definition. It is conjectured that this strong condition might be weakened for some other means, too.

Original languageEnglish
Pages (from-to)129-139
Number of pages11
JournalAnnales Mathematicae et Informaticae
Volume32
Publication statusPublished - Jan 1 2005

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Keywords

  • Geometric mean
  • Information geometry
  • Logarithmic mean
  • Operator means
  • Positive matrices

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)

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