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Arithmetic-geometric means of positive matrices

Published online by Cambridge University Press:  24 October 2008

Joel E. Cohen  and
Roger D. Nussbaum
Joel E. Cohen
Affiliation:
Rockefeller University, New York, NY 10021, U.S.A.
Roger D. Nussbaum
Affiliation:
Rutgers University, New Brunswick, NJ 08903, U.S.A.
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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 = (aij) with positive elements aij, define (contrary to custom) A½ elementwise by [A½]ij = (aij)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−1A(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).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 209 - 219
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

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