mean and variance normalization of vectors

I have vectors and I expect some multivariate normal distribution.

I want to normalize the vectors in such a way that y = M (x - b) has mean zero (E[Y] = 0) and variance 1 (var(Y) = 1).

For n=1, that is basically the Standard score.

1. With M = 1 and b = E[X], I get it mean-normalized but not variance-normalized.

2. With M being a non-negative symmetric square root of var(X)^{-1}, we get var(Y) = 1. We can choose b as in case 1.

I want to estimate M and b based on some running/moving algorithm. For the first case, this is easy:

For every new x, I can use the update rule

for some α. I could set α = 1/N and set N ← N + 1 in every iteration, starting with N = 1.

How would I do something similar for the second case, esp. for M?

I could use a similar update rule to calculate the covariance matrix, e.g.

but I'm not sure how well that works in practice (also, I haven't really seen that formula in such form) or if that even converges (given that it uses also the moving b).

Also, if I use this, I only have an estimation of var(X) but I still need the M. I probably could use another estimation to get the M based on the Σ (which one?), but I wonder if I could just get an estimation of M more directly.

(I asked the same question also on Math.StackExchange.)