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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

Cited in: Zbl pre06142087 Zbl pre06142085 Zbl 1235.65038 Zbl 1220.26024 Zbl 1217.65075 Zbl 1214.47016 Zbl 1194.65065 Zbl 1189.15030 Zbl 1186.47013 Zbl 1182.47021 Zbl 1182.47020 Zbl 1172.47017 Zbl 1160.47016 Zbl 1153.15021 Zbl 1132.47016 Zbl 1129.47017 Zbl 1126.47016 Zbl 1108.47020 Zbl 1083.15032

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