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Zbl 1063.47013
Ando, T.; Li, Chi-Kwong; Mathias, Roy
Geometric means.
(English)
[J] Linear Algebra Appl. 385, 305-334 (2004). ISSN 0024-3795

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 $T(A)\equiv (G((A_i)_{i\ne 1}), G((A_i)_{i\ne 2}),\dots,G((A_i)_{i\ne k+1}))$. \par 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.
[Yiu Tung Poon (Ames)]
MSC 2000:
*47A64 Operator means etc.
47A63 Operator inequalities, etc.
15A45 Miscellaneous inequalities involving matrices

Keywords: positive semidefinite matrix; geometric mean; matrix square root; matrix inequality; spectral radius

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