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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 AB=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=XY 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+2BX -1 B, Y=X-BY -1 B. He computes the explicit solution T:=(1/2)(A+A(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 2BT, i.e. BA.

MSC:
15A24Matrix equations and identities
15A48Positive matrices and their generalizations (MSC2000)
15A29Inverse problems in matrix theory