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Let $A,B$ be positive definite (Hermitian) matrices. Define the Arithmetic-geometric means of positive matrices by $A_0=A, G_0=B$, $A_{n+1}=\frac{A_n+G_n}{2}, G_{n+1}=A_n\natural G_n$, where $A_n\natural G_n$ means the geometric mean defined inhttp://www.isid.ac.in/~statmath/eprints/2011/isid201102.pdfin http://www.isid.ac.in/~statmath/eprints/2011/isid201102.pdf

Will ${A_n}$ and ${G_n}$ converge to the same matrix?

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# Arithmetic-geometric mean of positive matrices

Let $A,B$ be positive definite (Hermitian) matrices. Define the Arithmetic-geometric means of positive matrices by $A_0=A, G_0=B$, $A_{n+1}=\frac{A_n+G_n}{2}, G_{n+1}=A_n\natural G_n$, where $A_n\natural G_n$ means the geometric mean defined inhttp://www.isid.ac.in/~statmath/eprints/2011/isid201102.pdf

Will ${A_n}$ and ${G_n}$ converge to the same matrix?