I'd like to find the correlation matrix of $Y_1$ and $Y_2$, for $Y_1=X_1+X_2$ and $Y_2=X_1-X_2$, where $X_1$ and $X_2$ are random variable with correlation matrix $\Omega_X=\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix} $, where $-1\leq \rho \leq 1$.
Here is my what I did:
I think that since the main diagonal entries are equal to 1, $\Omega_X=\Sigma_X$. Then, I apply the Cholesky decomposition: $\Sigma_X=U^t\,U$
$\begin{pmatrix} X_1\\ X_2 \end{pmatrix} = U^t \, \begin{pmatrix} Z_1\\ Z_2 \end{pmatrix}$, $Z_1$ and $Z_2$ are independent standard normal variables.
$\Rightarrow \begin{pmatrix} X_1\\ X_2 \end{pmatrix} = \begin{pmatrix} 1 & 0\\ \rho & \sqrt{1-\rho^2} \end{pmatrix} \, \begin{pmatrix} Z_1\\ Z_2 \end{pmatrix} $
$\Rightarrow \begin{cases} X_1 = Z_1\\ X_2 = \rho\,Z_1 + \sqrt{1-\rho^2}Z_2 \end{cases} $
$ \Rightarrow \begin{cases} Y_1 = (1+\rho)\,Z_1 + \sqrt{1-\rho^2}\,Z_2\\ Y_2 = (1-\rho)\,Z_1 - \sqrt{1-\rho^2}\,Z_2 \end{cases} $
I deduce: $Y_1 \sim N(0,2\rho+2)$ and $Y_2 \sim N(0,2-2\rho)$
But I am not able to get corr($Y_1$,$Y_2$).
Any hint about what I'm doing wrong? Thank you
I don't see anything about normal variables here, so Cholesky is out.
Also I don't see any assumptions about the variance of $X_1$ and $X_2.$ I will call them $\sigma_1^2$ and $\sigma_2^2$ so that the covariance matrix of $X_1$ and $X_2$ is $\begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2\\\rho\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix}.$ The final answer depends on these variances (for instance, if they're equal then the correlation of $Y_1$ and $Y_2$ is zero).
First get variance of $Y_1$ and $Y_2.$ $$ Var(Y_1) = Var(X_1) + Var(X_2) + 2Cov(X_1,X_2) = \sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2$$ and similarly $$ Var(Y_2) = \sigma_1^2+\sigma^2_2-2\rho\sigma_1\sigma_2.$$
Then you can get the covariance of $Y_1$ and $Y_2$ by $$ Cov(Y_1,Y_2) = E(Y_1Y_2)-E(Y_1)E(Y_2) = E(X_1^2-X_2^2)-(E(X_1)^2-E(X_2)^2) \\= Var(X_1)-Var(X_2) = \sigma_1^2-\sigma_2^2.$$ Then the correlation of $Y_1$ and $Y_2$ is $$ Cor(Y_1,Y_2) = \frac{Cov(Y_1,Y_2)}{\sqrt{Var(Y_1)Var(Y_2)}}$$