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A note on computing matrix geometric means. (English) Zbl 1293.65068

Summary: A new definition is introduced for the matrix geometric mean of a set of \(k\) positive definite \(n\times n\) matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires \(O(n^3 k^2)\) arithmetic operations per step whereas the methods based on the symmetrization technique of T. Ando, C.-K. Li and R. Mathias [Linear Algebra Appl. 385, 305–334 (2004; Zbl 1063.47013)] have complexity \(O(n^3 k!2^k)\). The new mean is obtained from the properties of the centroid of a triangle rephrased in terms of geodesics in a suitable Riemannian geometry on the set of positive definite matrices. It satisfies most part of the ten properties stated by Ando et al.; a counterexample shows that monotonicity is not fulfilled.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A15 Determinants, permanents, traces, other special matrix functions
15B48 Positive matrices and their generalizations; cones of matrices

Citations:

Zbl 1063.47013
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Full Text: DOI

References:

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