*(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 $A\u266fB={A}^{1/2}{\left({A}^{-1/2}B{A}^{-1/2}\right)}^{1/2}{A}^{1/2}$.

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 $A=C(X,Y)$, $B=X\u266fY$ 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 $X=A+2B{X}^{-1}B$, $Y=X-B{Y}^{-1}B$. He computes the explicit solution $T:=(1/2)(A+A\u266f(A+8B{A}^{-1}B))$ 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 |