Multiplying a vector of independant gaussian r.v. by an orthogonal matrix gives independant r.v.

Question

I found in a proof (about $\chi^2$ and Student laws) :

Let $$\begin{pmatrix}V_1 \\ V_2 \\ ... \\ V_n \end{pmatrix} = A \begin{pmatrix}Z_1 \\ Z_2 \\ ... \\ Z_n \end{pmatrix}$$

Since $Z_i$ are independent gaussian random variables and A is an orthogonal matrix, the $V_i$ are independent random variables as well.

How can we prove this, using the orthogonality of A ?

Answer 1

The $V_i$ are jointly Gaussian random variables with covariance matrix $\hat{C} = ACA^T$ where $C$, the covariance matrix of the $Z_i$, is a diagonal matrix since the $Z_i$ are given to be independent. So, if you can show that $\hat{C}$ is also a diagonal matrix, you will have proved that the $V_i$ are independent random variables.