AbstractWe 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).