Given a set of normally distributed random variables $_ ( = 1,2, ⋯ , )$ with covariance matrix = $_{}$ , assuming all the $_ ( = 1,2, ⋯ , )$ have 0 mean.
Please outline an algorithm to transform $_ , = 1,2, ⋯ , $ into a new set of random variables $_ ( = 1,2, ⋯ , )$ using linear transformation, i.e, please find coefficients ${} such that
$$ = \sum_{i=1}^n _{}_{} ,= 1 , 2 , ⋯ $$ Where $_$ and $_$ have 0 correlation for every$ , = 1,2,⋯, ≠ $.
Assume n=2
$$c_{ij}= \begin{pmatrix} Cov(X_1,X_1) & Cov(X_1,X_2) \\ Cov(X_2, X_1) & Cov(X_2,X_2) \\ \end{pmatrix} $$
$$Y_1=b_{11}X_1+b_{21}X_2$$ $$ Y_2=b_{12}X_1+b_{22}X_2$$
Since Correlation=$\frac{Cov}{{\sigma_X}{\sigma_Y}}$,then try to get Cov=0.
By the linear relationship,
$$Cov(Y_1,Y_2)=Cov(b_{11}X_1,b_{12}X_1)+Cov(b_{11}X_1,b_{22}X_2)+Cov(b_{21}X_2,b_{12}X_2)+Cov(b_{21}X_2,b_{22}X_2)$$
$$Cov(Y_1,Y_2)=b_{11}b_{12}Cov(X_1,X_1)+b_{11}b_{22}Cov(X_1,X_2)+b_{21}b_{12}Cov(X_2,X_2)+b_{21}b_{22}Cov(X_2,X_2)$$
Try to diagonalize the Covariance Matrix，
$$\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \begin{pmatrix} Cov(X_1,X_1) & Cov(X_1,X_2) \\ Cov(X_2, X_1) & Cov(X_2,X_2) \\ \end{pmatrix} \begin{pmatrix} e & f \\ g & h \\ \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} $$
Then, $$ecCov(X_1,X_1)+edCov(X_2,X_1)+gcCov(X_1,X_2)+gdCov(X_2,X_2)=0$$ the corresponding coefficient is $$b_{11}=e,b_{12}=c,b_{21}=g,b_{22}=d$$