I have a couple questions (maybe all related) and I think they are because my matrix math is weak / rusty, weak and rusty

1) Why is Z = (Covariance Matrix)^(-1/2) * (X_vector - mu_vector) N_p (0,I)?

2) To say Z^2 does not mean (Covariance Matrix)^(-1/2) * (X_vector - mu_vector) * (Covariance Matrix)^(-1/2) * (X_vector - mu_vector). Instead the order changes to:

(X_vector - mu_vector)' * (Covariance Matrix)^(-1/2)*(Covariance Matrix)^(-1/2)* (X_vector - mu_vector).

I understand that for the calculations to work, we need to have the dimensions work, but why is the order the way it is (and how do you know?)

3) Finally trying to show sqrt(n) * (Covariance Matrix)^(-1/2) * (X_vector - mu_vector) is N_p (0,I).

I bet all these show the same deficiency I have but if anyone can help explain I would be grateful. Thanks!

Brian