A covariance matrix naturally transforms from the measured space to a space where things are approximately unit Gaussian distributed. This is identical to the Z transform in 1D case.
This can be useful in, say, exotic options trading - a natural unit of measurement is how many ‘vols’ an underlier has moved, e.g. a 10-vol move is very large.
Not really the covariance matrix, though, but its Cholesky decomposition (which exists, as a covariance matrix is symmetric positive (semi)definite, as otherwise you could construct a linear combination with negative variance).
Useful stuff.
And vice versa, btw - take iid RV with unit variance, hit them with the Cholesky decomposition, and you have the desired covariance. Used all over Monte Carlo and finance and so on.
This can be useful in, say, exotic options trading - a natural unit of measurement is how many ‘vols’ an underlier has moved, e.g. a 10-vol move is very large.