“Heinz mean”的意思、由来-开放百科全书

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:

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 < {{sfrac|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. ^{{citation|first=R.|last=Bhatia|title=Interpolating the arithmetic-geometric mean inequality and its operator version|journal=Linear Algebra and its Applications|volume=413|issue=2–3|pages=355–363|year=2006|doi=10.1016/j.laa.2005.03.005}}.
3. ^{{citation|first1=R.|last1=Bhatia|first2=C.|last2=Davis|authorlink2=Chandler Davis|title=More matrix forms of the arithmetic-geometric mean inequality|journal=SIAM Journal on Matrix Analysis and Applications|volume=14|issue=1|pages=132–136|year=1993|doi=10.1137/0614012}}.
4. ^{{citation|first=Koenraad M.R.|last=Audenaert|title=A singular value inequality for Heinz means|arxiv=math/0609130 |journal=Linear Algebra and its Applications|volume=422|issue=1|pages=279–283|year=2007|doi=10.1016/j.laa.2006.10.006}}.