## Abstract

We prove the existence of unique limits and establish inequalities for matrix generalizations of the arithmetic-geometric mean of Lagrange and Gauss. For example, for a matrix A = (a_{ij}) with positive elements a_{ij}, define (contrary to custom) Ai elementwise by [A^{½}]ij = (a_{ij})^{½} Let A(0) and B(0) be d×d matrices (1 < d < ∞) with all elements positive real numbers. Let A(n+1) = (A(n) + B(n))/2 and B(n+ 1) = (d^{−1}A(n) B(n)) ½. Then all elements of A(n) and B(n) approach a common positive limit L. When A(0) and B(0) are both row-stochastic or both column-stochastic, dL is less than or equal to the arithmetic average of the spectral radii of A(0) and B(0).

Original language | English (US) |
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Pages (from-to) | 209-219 |

Number of pages | 11 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 101 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1987 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)