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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 {\it W. N. Anderson jun., M. E. Mays, T. D. Morley} and {\it 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. $B\leq A$.

15A24Matrix equations and identities
15B48Positive matrices and their generalizations; cones of matrices
15A29Inverse problems in matrix theory
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. Graduate texts in mathematics 191 (1999)
[17] Liu, X.; Gao, H.: On the positive definite solutions of the matrix equations $Xs{\pm}$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 $X+A*F(X)$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