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Geometric means. (English) Zbl 1063.47013
Let $G(A,B)$ be the geometric mean of two $n\times n$ positive semidefinite matrices $A$ and $B$. The authors extend the definition of $G$ to any number of $n\times n$ positive semidefinite matrices inductively. Suppose that for some $k\ge 2$, the geometric mean $G(A_1,A_2,\dots,A_k)$ of any $k$ positive semidefinite matrices $A_1,A_2,\dots,A_k$ has been defined. Let $A=(A_1,A_2,\dots,A_k,A_{k+1})$ be a $(k+1)$-tuple of $n\times n$ positive semidefinite matrices. Define $T(A)\equiv (G((A_i)_{i\ne 1}), G((A_i)_{i\ne 2}),\dots,G((A_i)_{i\ne k+1}))$. The authors show that the sequence $(T^r(A))_{r=1}^{\infty}$ has a limit of the form $(\tilde A,\dots,\tilde A)$ and define $G(A_1,A_2,\dots,A_k,A_{k+1})=\tilde A$. The definition given here is the only one in the literature that has the properties that one would expect from a geometric mean. The authors also prove some new properties of the geometric mean of two matrices, and give some simple computational formulae related to them for $2\times 2$ matrices.

47A64Operator means, shorted operators, etc.
47A63Operator inequalities
15A45Miscellaneous inequalities involving matrices
Full Text: DOI
[1] Jr., W. N. Anderson; Duffin, R. J.: Series and parallel addition of matrices. J. math. Anal. appl. 26, 576-594 (1969) · Zbl 0177.04904
[2] Jr., W. N. Anderson; Morley, T. D.; Trapp, G. E.: Symmetric function means of positive operators. Linear algebra appl. 60, 129-143 (1984) · Zbl 0575.47001
[3] Choi, M. -D.: Some assorted inequalities for positive linear maps on C*-algebras. J. operator theory 4, 271-285 (1980) · Zbl 0511.46051
[4] P. Dukes, M.-D. Choi, A geometric mean of three positive matrices, 1998
[5] Golub, G. H.; Welsch, J. F.: Calculation of Gaussian quadrature rules. Math. comp. 23, 221-230 (1969) · Zbl 0179.21901
[6] Higham, N. J.: Accuracy and stability of numerical algorithms. (1996) · Zbl 0847.65010
[7] H. Kosaki, Geometric mean of several positive operators, 1984 · Zbl 0577.47041
[8] Li, C. -K.; Mathias, R.: Extremal characterizations of the Schur complement and resulting inequalities. SIAM rev. 42, No. 2, 233-246 (2000) · Zbl 0948.15017
[9] Trapp, G. E.: Hermitian semidefinite matrix means and related matrix inequalities----an introduction. Linear multilinear algebra 16, 113-123 (1984) · Zbl 0548.15013