Let $X = (X_1,...,X_n)$ and $Y=(Y_1,...,Y_m)$ be two random vectors. Let $C_{XY}$ be the covariance matrix of two random vectors $X$ and $Y$.
What is the interpretation of the matrix $A = C_{XX}^{-1/2} C_{XY} C_{YY}^{-1/2}$?
It seems to me like it is very much related to the correlation matrix, but it is not exactly a correlation matrix (because its elements are actually not $Corr(X_i,X_j)$).
However, if $m=n=1$, then we get that $A$ is exactly a 1 x 1 matrix that corresponds to the correlation between $X$ and $Y$.