Proof of $\operatorname{Corr}\left(X, Y\middle|Z\right) = \operatorname{Corr}(X-\hat X, Y-\hat Y)$ The text I'm reading claims that for multivariate $X$, $Y$, and $Z$, and $\hat X$ and $\hat Y$ are the regression of $X$ and $Y$ on $Z$ we have that

$$\operatorname{Corr}\left(X, Y\middle|Z\right) = \operatorname{Corr}(X-\hat X, Y-\hat Y)$$

However, there was no derivation of this claim. What steps are required to show this identity, and why is the assumption of normal variables required?
 A: This is definitional and comes from the fact that $$X_i = Z_i\beta_x + \epsilon_{xi}$$, $$\hat{X}_i = Z_i\beta_x$$ and $$X_i|Z_i = \epsilon_{xi} = X_i - Z_i\beta_x = X_i - \hat{X}_i$$ (and the same for $Y$). These all come from the definition of regression. Then you can simply substitute: $$Corr(X,Y|Z) = Corr(X|Z, Y|Z) = Corr(X - \hat{X}, Y - \hat{Y})$$.
A: For a multivariate normal random vector partitioned as $\mathbf{x}=\left[\begin{matrix}\mathbf{x}_1 \\ \mathbf{x}_2\end{matrix}\right]$, the conditional distribution of $\mathbf{x}_1$ conditional on on $\mathbf{x}_2$ is also multivariate normal with mean vector equal to the general expression for the best linear minimum mean square error predictor $\hat{\mathbf{x}}_1$ of $\mathbf{x}_1$ based on $\mathbf{x}_2$ (applying to any multivariate distribution) and, importantly, a variance-covariance matrix that does not depend on $\mathbf{x}_2$.
In the multivariate normal case we can thus write
$$
\mathbf{x}_1=\hat{\mathbf{x}}_1 + \mathbf{e}_1,
$$ 
where $\mathbf{e}_1$ is independent of $\hat{\mathbf{x}}_1$ and multivariate normal with zero mean and variance-covariance matrix equal to the the above conditional variance-covariance matrix.
Hence,
$$
\operatorname{Var}(\mathbf{x}_1-\hat{\mathbf{x}}_1) = \operatorname{Var}\mathbf{e}_1=\operatorname{Var}(\mathbf{x}_1|\mathbf{x}_2),
$$
that is, all variances, covariances and correlations of deviations from $\hat{\mathbf{x}}_1$ equals the corresponding conditional quantities.
This equality does not hold in general, however, as we can easily construct a multivariate distribution for which the conditional variance-covariance matrix of $\mathbf{x}_1$ depends on $\mathbf{x}_2$.
