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Zbl 0641.15009
Anderson, William N.jun.; Mays, Michael E.; Morley, Thomas D.; Trapp, George E.
The contraharmonic mean of HSD matrices. (English)
[J] SIAM J. Algebraic Discrete Methods 8, 674-682 (1987). ISSN 0196-5212

The contraharmonic mean of two positive semidefinite Hermitian matrices A and B is defined by the relation $C(A,B)=A+B-2(A:B),$ where $A:B=A(A+B)\sp{-1}B$ is the so-called parallel addition introduced by {\it W. N. Anderson} jun. and {\it R. J. Duffin} [J. Math. Anal. Appl. 26, 576-594 (1969; Zbl 0177.049)]. The dual of the contraharmonic mean of A and B is given by $C'(A,B)=C(A\sp{-1},B\sp{-1})\sp{-1}.$ It is shown that $$ C'(A,B)=A:B+2(A:B)C(A,B)\sp{-1}(A:B)=(A(A:B)\sp{- 1}A):(B(A:B)\sp{-1}B). $$ With the aid of the contraharmonic mean and its dual the authors study fixed point problems, the monotonicity behaviour of C(A,B), an infinite family of means for positive semidefinite Hermitian matrices that generalize C(A,B), inverse mean problems, and connections between C(A,B) and least square problems.
[A.R.Kräuter]

MSC 2000:: *15A45 Miscellaneous inequalities involving matrices
15A24 Matrix equations
15A27 Commutativity of matrices

Keywords: parallel addition of matrices; contraharmonic mean; positive semidefinite Hermitian matrices; fixed point problems; monotonicity behaviour; inverse mean problems; least square problems

Citations: Zbl 0177.049

Cited in: Zbl 1096.15005

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