Geometric means
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 \(k\geq 2\), 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 \(T(A)\equiv (G((A_i)_{i\neq 1}), G((A_i)_{i\neq 2}),\dots,G((A_i)_{i\neq k+1}))\). 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.
scientific article; zbMATH DE number 2083518