The contraharmonic mean of the positive definite matrices and is defined by . (It generalizes the contraharmonic mean of scalars .) Their geometric mean is defined by .
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 and for the system of nonlinear matrix equations , where and are given positive definite -matrices.
The author shows that the IMP is equivalent to solving the system of well-known matrix equations , . He computes the explicit solution to the first equation and then solves the second equation with . The IMP is solvable if and only if , i.e. .