 To see that formula_128 is independent of formula_31 remark first that this is the case for any diagonal matrix formula_31, because the coordinate-wise maximizer is independent of a scaling factor.Then, rotation of the data points or choosing formula_31 non-diagonal are equivalent.The rank-formula_100 update of the covariance matrix, that is, the right most summand in the update equation of formula_79, maximizes a log-likelihood in that

for formula_135 (otherwise formula_31 is singular, but substantially the same result holds for formula_137).