Geometric means of structured matrices

Abstract

The geometric mean of positive definite matrices is usually identified with the Karcher mean, which possesses all properties—generalized from the scalar case—a geometric mean is expected to satisfy. Unfortunately, the Karcher mean is typically not structure preserving, and destroys, e.g., Toeplitz and band structures, which emerge in many applications. For this reason, the Karcher mean is not always recommended for modeling averages of structured matrices. In this article a new definition of a geometric mean for structured matrices is introduced, its properties are outlined, algorithms for its computation, and numerical experiments are provided. In the Toeplitz case an existing mean based on the Kähler metric is analyzed for comparison.

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Acknowledgements

The authors would like to thank the referees for many insightful comments which improved the presentations of the paper. In particular, they are indebted with a referee which provided the elegant proof for Theorem 2.1.

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Correspondence to Raf Vandebril.

Additional information

This work was partially supported by MIUR grant number 2002014121; by the Research Council KU Leuven, projects OT/11/055 (Spectral Properties of Perturbed Normal Matrices and their Applications), CoE EF/05/006 Optimization in Engineering (OPTEC); by the Fund for Scientific Research–Flanders (Belgium) project G034212N (Reestablishing Smoothness for Matrix Manifold Optimization via Resolution of Singularities); and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization).

Communicated by Peter Benner.

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Bini, D.A., Iannazzo, B., Jeuris, B. et al. Geometric means of structured matrices. Bit Numer Math 54, 55–83 (2014). https://doi.org/10.1007/s10543-013-0450-4

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Keywords

  • Matrix geometric mean
  • Structured matrices
  • Manifold optimization

Mathematics Subject Classification (2010)

  • 14Q10
  • 15A24
  • 15B05
  • 53B21
  • 65K10