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The inverse mean problem of geometric mean and contraharmonic means. (English) Zbl 1096.15005

The contraharmonic mean $C\left(A,B\right)$ of the positive definite matrices $A$ and $B$ is defined by $C\left(A,B\right)=A+B-2{\left({A}^{-1}+{B}^{-1}\right)}^{-1}$. (It generalizes the contraharmonic mean of scalars $\left({a}^{2}+{b}^{2}\right)/\left(a+b\right)$.) Their geometric mean is defined by $A♯B={A}^{1/2}{\left({A}^{-1/2}B{A}^{-1/2}\right)}^{1/2}{A}^{1/2}$.

The inverse mean problem (IMP) of contraharmonic and geometric means [proposed in W. N. Anderson jun., M. E. Mays, T. D. Morley and 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 $A=C\left(X,Y\right)$, $B=X♯Y$ where $A$ and $B$ are given positive definite $n×n$-matrices.

The author shows that the IMP is equivalent to solving the system of well-known matrix equations $X=A+2B{X}^{-1}B$, $Y=X-B{Y}^{-1}B$. He computes the explicit solution $T:=\left(1/2\right)\left(A+A♯\left(A+8B{A}^{-1}B\right)\right)$ to the first equation and then solves the second equation with $X=T$. The IMP is solvable if and only if $2B\le T$, i.e. $B\le A$.

##### MSC:
 15A24 Matrix equations and identities 15A48 Positive matrices and their generalizations (MSC2000) 15A29 Inverse problems in matrix theory