Let \[ \begin{pmatrix}X_1\\X_2 \end{pmatrix}\sim N\left( {\mu_1\choose \mu_2}, \begin{pmatrix}\Sigma_{11}& \Sigma_{12}\\\Sigma_{21}& \Sigma_{22} \end{pmatrix}\right). \] Then, \(X_1\) and \(X_2\) are independent \(\iff\) \(\Sigma_{12}=\Sigma_{21}=0\).
If \(X\sim N_p(\mu, \Sigma)\) and \(A\Sigma B'=0\), then \(AX\) and \(BX\) are independent.
\({AX \choose BX}\sim N \left( \begin{pmatrix} A\mu \\ B\mu \end{pmatrix}, \begin{pmatrix} A\Sigma A'& A\Sigma B'\\ (A\Sigma B')'& B\Sigma B'\end{pmatrix} \right)\).
\(\text{Cov}(AX,BX)= A\Sigma B'=0\iff AX\perp BX.\)
확장: If \(X\sim N_p(\mu, \Sigma)\) and \(A_1,\ldots,A_k\) saitsfy \(A_i\Sigma A_j'=0\) for \(i\ne j\), then \(A_1X\) and \(A_kX\) are independent.
Suppose \(Y\sim N(\mu,\sigma^2I).\) If \(P_1,\ldots,P_k\) are orthogonal projections satisfying \(P_iP_j=0\) for \(i\ne j\), then
\(P_1Y,P_2Y,\ldots ,P_kY\) are independent;
\(||P_1Y||^2,\ldots,||P_kY||^2\) are independent;
\(P_iY\sim N(P_i\mu, \sigma^2P_i)\).