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( T o p )
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P r i n t / e x p o r t
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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
e
In mathematics, the Heinz mean (named after E. Heinz [1] real numbers A B [2]
H
x
(
A ,
B )
=
A
x
B
1 −
x
+
A
1 −
x
B
x
2
,
{\displaystyle \operatorname {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 arithmetic (x geometric (x x 1 / 2
A B
=
H
1 2
(
A ,
B )
<
H
x
(
A ,
B )
<
H
0
(
A ,
B )
=
A +
B
2
.
{\displaystyle {\sqrt {AB}}=\operatorname {H} _{\frac {1}{2}}(A,B)<\operatorname {H} _{x}(A,B)<\operatorname {H} _{0}(A,B)={\frac {A+B}{2}}.}
The Heinz means appear naturally when symmetrizing
α
{\textstyle \alpha }
[3]
It may also be defined in the same way for positive semidefinite matrices , and satisfies a similar interpolation formula.[4] [5]
See also [ edit ]
References [ edit ]
^ E. Heinz (1951), "Beiträge zur Störungstheorie der Spektralzerlegung", Math. Ann. , 123 , pp. 415–438.
^ 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
^
Nielsen, Frank; Nock, Richard; Amari, Shun-ichi (2014), "On Clustering Histograms with k-Means by Using Mixed α-Divergences", Entropy , 16 6 ): 3273–3301, Bibcode :2014Entrp..16.3273N , doi :10.3390/e16063273 hdl :1885/98885
^ 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
^ Audenaert, Koenraad M.R. (2007), "A singular value inequality for Heinz means", Linear Algebra and Its Applications , 422 (1 ): 279–283, arXiv :math/0609130 doi :10.1016/j.laa.2006.10.006 , S2CID 15032884
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Heinz_mean&oldid=1170891817 " C a t e g o r i e s : ● A p p l i e d m a t h e m a t i c s s t u b s ● M e a n s H i d d e n c a t e g o r i e s : ● A r t i c l e s w i t h s h o r t d e s c r i p t i o n ● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a ● A l l s t u b a r t i c l e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 7 A u g u s t 2 0 2 3 , a t 2 0 : 3 6 ( U T C ) . ● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n . ● P r i v a c y p o l i c y ● A b o u t W i k i p e d i a ● D i s c l a i m e r s ● C o n t a c t W i k i p e d i a ● C o d e o f C o n d u c t ● D e v e l o p e r s ● S t a t i s t i c s ● C o o k i e s t a t e m e n t ● M o b i l e v i e w ●
T o g g l e l i m i t e d c o n t e n t w i d t h
