Heinz mean

In mathematics, the Heinz mean (named after E. Heinz[1]) of two non-negative real numbers A and B, was defined by Bhatia[2] as:

with 0 ≤ x ≤ 1/2.

For different values of x, this Heinz mean interpolates between the arithmetic (x = 0) and geometric (x = 1/2) means such that for 0 < x < 1/2:

The Heinz mean may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.[3][4]

See also

References

  1. ↑ E. Heinz (1951), "Beiträge zur Störungstheorie der Spektralzerlegung", Math. Ann., 123, pp. 415–438.
  2. ↑ Bhatia, R. (2006), "Interpolating the arithmetic-geometric mean inequality and its operator version", Linear Algebra and its Applications, 413 (2–3): 355–363, doi:10.1016/j.laa.2005.03.005.
  3. ↑ Bhatia, R.; Davis, C. (1993), "More matrix forms of the arithmetic-geometric mean inequality", SIAM Journal on Matrix Analysis and Applications, 14 (1): 132–136, doi:10.1137/0614012.
  4. ↑ Audenaert, Koenraad M.R. (2007), "A singular value inequality for Heinz means", Linear Algebra and its Applications, 422 (1): 279–283, arXiv:math/0609130Freely accessible, doi:10.1016/j.laa.2006.10.006.


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