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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 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 $$A=C(X,Y)$$, $$B=X\sharp Y$$ 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:
 15A24 Matrix equations and identities 15B48 Positive matrices and their generalizations; cones of matrices 15A29 Inverse problems in linear algebra
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