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The inverse mean problem of geometric mean and contraharmonic means. (English) Zbl 1096.15005
The contraharmonic mean $C(A,B)$ of the positive definite matrices $A$ and $B$ is defined by $C(A,B)=A+B-2(A^{-1}+B^{-1})^{-1}$. (It generalizes the contraharmonic mean of scalars $(a^2+b^2)/(a+b)$.) Their geometric mean is defined by $A\sharp B= A^{1/2}(A^{-1/2} BA^{-1/2})^{1/2}A^{1/2}$. The inverse mean problem (IMP) of contraharmonic and geometric means [proposed in {\it W. N. Anderson jun., M. E. Mays, T. D. Morley} and {\it G. E. Trapp}, SIAM J. Algebraic Discrete Methods 8, 674--682 (1987; Zbl 0641.15009)] is to find positive definite matrices $X$ and $Y$ for the system of nonlinear matrix equations $A=C(X,Y)$, $B=X\sharp Y$ where $A$ and $B$ are given positive definite $n\times n$-matrices. The author shows that the IMP is equivalent to solving the system of well-known matrix equations $X=A+2BX^{-1}B$, $Y=X-BY^{-1}B$. He computes the explicit solution $T:=(1/2)(A+A\sharp (A+8BA^{-1}B))$ to the first equation and then solves the second equation with $X=T$. The IMP is solvable if and only if $2B\leq T$, i.e. $B\leq A$.

MSC:
15A24Matrix equations and identities
15B48Positive matrices and their generalizations; cones of matrices
15A29Inverse problems in matrix theory
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References:
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