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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 $Y$.

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The variance-covariance matrix of $\tilde X=C_{XX}^{-1/2}X$ is the identity matrix $I_n$. The variance-covariance matrix of $\tilde Y=C_{YY}^{-1/2}Y$ is the identity matrix $I_m$. The covariance matrix of $\tilde X$ and $\tilde Y$ is the matrix $A$.

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