# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Geometric means. (English) Zbl 1063.47013

Let $G\left(A,B\right)$ be the geometric mean of two $n×n$ positive semidefinite matrices $A$ and $B$. The authors extend the definition of $G$ to any number of $n×n$ positive semidefinite matrices inductively. Suppose that for some $k\ge 2$, the geometric mean $G\left({A}_{1},{A}_{2},\cdots ,{A}_{k}\right)$ of any $k$ positive semidefinite matrices ${A}_{1},{A}_{2},\cdots ,{A}_{k}$ has been defined. Let $A=\left({A}_{1},{A}_{2},\cdots ,{A}_{k},{A}_{k+1}\right)$ be a $\left(k+1\right)$-tuple of $n×n$ positive semidefinite matrices. Define $T\left(A\right)\equiv \left(G\left({\left({A}_{i}\right)}_{i\ne 1}\right),G\left({\left({A}_{i}\right)}_{i\ne 2}\right),\cdots ,G\left({\left({A}_{i}\right)}_{i\ne k+1}\right)\right)$.

The authors show that the sequence ${\left({T}^{r}\left(A\right)\right)}_{r=1}^{\infty }$ has a limit of the form $\left(\stackrel{˜}{A},\cdots ,\stackrel{˜}{A}\right)$ and define $G\left({A}_{1},{A}_{2},\cdots ,{A}_{k},{A}_{k+1}\right)=\stackrel{˜}{A}$. 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 $2×2$ matrices.

##### MSC:
 47A64 Operator means, shorted operators, etc. 47A63 Operator inequalities 15A45 Miscellaneous inequalities involving matrices