Heinz mean

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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:

H_x(A, B) = \frac{A^x B^{1-x} + A^{1-x} B^x}{2}.

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:

 \sqrt{A B} = H_{1/2}(A, B) < H_x(A, B) < H_0(A, B) = \frac{A + B}{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.
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