  Here we prove the convergence of the Ando-Li-Mathias and Bini-Meini-Poloni
procedures for matrix means. Actually it is proved here that for a two-variable
function which maps pairs of positive definite matrices to a positive definite
matrix and is not greater than the square mean of two positive definite
matrices, the Ando-Li-Mathias and Bini-Meini-Poloni procedure converges. In
order to be able to set up the Bini-Meini-Poloni procedure, a weighted
two-variable matrix mean is also needed. Therefore a definition of a
two-variable weighted matrix mean corresponding to every symmetric matrix mean
is also given. It is also shown here that most of the properties considered by
Ando, Li and Mathias for the $n$-variable geometric mean hold for all of these
$n$-variable maps that we obtain by this two limiting process for all
two-variable matrix means. As a consequence it also follows that the
Bini-Meini-Poloni procedure converges cubically for every matrix mean.
