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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. \(B\leq A\).

15A24 Matrix equations and identities
15B48 Positive matrices and their generalizations; cones of matrices
15A29 Inverse problems in linear algebra
Full Text: DOI
[1] Anderson, W.N.; Morley, T.D.; Trapp, G.E., Positive solutions to X=A−BX−1B*, Linear algebra appl., 134, 53-62, (1990) · Zbl 0702.15009
[2] Anderson, W.N.; Mays, M.E.; Morley, T.D.; Trapp, G.E., The contraharmonic Mean of HSD matrices, SIAM J. algebra discr. meth., 8, 674-682, (1987) · Zbl 0641.15009
[3] Anderson, W.N.; Trapp, G.E., Inverse problems for means of matrices, SIAM J. algebra discr. meth., 7, 188-192, (1986) · Zbl 0596.15010
[4] T. Ando, Topics on Operator Inequalities, Lecture Notes Hokkaido University, Sapporo, 1978. · Zbl 0388.47024
[5] Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear algebra appl., 26, 203-241, (1979) · Zbl 0495.15018
[6] Ando, T., On the arithmetic-geometric-harmonic-Mean inequalities for positive definite matrices, Linear algebra appl., 52/53, 31-37, (1983) · Zbl 0516.15011
[7] Ando, T.; Li, C.-H.; Mathias, R., Geometric means, Linear algebra appl., 385, 305-334, (2004) · Zbl 1063.47013
[8] Corach, G.; Porta, H.; Recht, L., Geodesics and operator means in the space of positive operators, Int. J. math., 4, 193-202, (1993) · Zbl 0809.47017
[9] El-sayed, S.M.; Ran, A.C.M., On an iteration method for solving a class of nonlinear matrix equations, SIAM J. matrix anal. appl., 23, 632-645, (2002) · Zbl 1002.65061
[10] Engwerda, J.C., On the existence of a positive definite solution of the matrix equation X+ATX−1A=I, Linear algebra appl., 194, 91-108, (1993) · Zbl 0798.15013
[11] Engwerda, J.C.; Ran, A.C.M.; Rijkeboer, A.L., Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A*X−1A=Q, Linear algebra appl., 186, 255-275, (1993) · Zbl 0778.15008
[12] Ferrante, A.; Levy, B., Hermitian solutions of the equation X=Q+NX−1N*, Linear algebra appl., 247, 359-373, (1996) · Zbl 0876.15011
[13] Fiedler, M.; Pták, V., A new positive definite geometric Mean of two positive definite matrices, Linear algebra appl., 251, 1-20, (1997) · Zbl 0872.15014
[14] Guo, C.-H.; Lancaster, P., Iterative solution of two matrix equations, Math. comput., 68, 1589-1603, (1999) · Zbl 0940.65036
[15] Kubo, F.; Ando, T., Means of positive linear operators, Math. ann., 246, 205-224, (1980) · Zbl 0412.47013
[16] Lang, S., Fundamentals of differential geometry, () · Zbl 0932.53001
[17] Liu, X.; Gao, H., On the positive definite solutions of the matrix equations Xs±ATX−ta=in, Linear algebra appl., 368, 83-97, (2003)
[18] Lawson, J.D.; Lim, Y., The geometric Mean, matrices, metrics, and more, Am. math. monthly, 108, 797-812, (2001) · Zbl 1040.15016
[19] Meini, B., Efficient computation of the extreme solutions of X+A*X−1A=Q and X−A*X−1A=Q, Math. comput., 71, 1189-1204, (2002) · Zbl 0994.65046
[20] Nesterov, Yu.E.; Todd, M.J., Self-scaled barriers and interior-point methods for convex programming, Math. operat. res., 22, 1-42, (1997) · Zbl 0871.90064
[21] Ran, A.C.M.; Reurings, M.C.B., On the nonlinear matrix equation \(X + A^* \mathcal{F}(X) A = Q\): solutions and perturbation theory, Linear algebra appl., 346, 15-26, (2002) · Zbl 1086.15013
[22] Zhan, X.; Xie, J., On the matrix equation X+ATX−1A=I, Linear algebra appl., 247, 337-345, (1996) · Zbl 0863.15005
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