1. ## Covariance question

If Z₁ is Normal (μ₁,σ²), Z₂ is Normal (μ₂,σ²) and Cov(Z₁,Z₂)=ρσ²
For (X₁,X₂)=(Z₁-Z₂,Z₁+Z₂)
How can i find the corresponding covariance matrix?
I know the mean vector is

μ₁-μ₂
μ₁+μ₂

and with Cov(Z₁,Z₂)=ρσ² it is possible to find the covariance matrix for Z₁ and Z₂ but i am not sure how to find covariance matrix for X₁,X₂.

2. ## Re: Covariance question

Let C denote the covariance matrix of the random vector $Z = (Z_{1},Z_{2}) \ .$ The random vector X is a linear transformation of the random vector Z.

$X = AZ\ ,$

where

$A = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \ .$

The covariance matrix of X is similar to the covariance matrix of Z, i.e.,

$\text{Cov}(X) = ACA^{T}\ ,$

with $A^{T}$ denoting the transpose of the matrix A.