The inverse mean problem of geometric mean and contraharmonic means (Q2568371)

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 \textit{W. N. Anderson jun., M. E. Mays, T. D. Morley} and \textit{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. \(B\leq A\).