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) Ai 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).
|Original language||English (US)|
|Number of pages||11|
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|State||Published - Mar 1987|
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