Consider two $N \times N$ dimensional real matrices $A$ and $B$.
$A$ is a diagonal matrix with all non-zero elements taken from a real Gaussian distribution with mean $\mu = 0$ and variance $\sigma = \alpha$, where $\alpha$ is some positive real number.
$B$ is a symmetric matrix with all off-diagonal elements taken from a real Gaussian distribution with mean $\mu = 0$ and variance $\sigma = \beta$, where $\beta$ is again some positive real number. All the diagonal elements of $B$ are $0.$
Finally consider the sum of these two matrices $H = A + B$.
The matrix $H$ is heteroskedastic, since its' matrix elements have non-uniform variance.
The question is to find the variance of the diagonal and off-diagonal elements of $H$ after some arbitrary orthogonal transformation:
$H \rightarrow \tilde{H} = C^{T}H C$
where $C$ is an orthogonal matrix.
For the purpose of this calculation, the following can be assumed, if needed:
- We are working in the large $-N$ limit.
- The matrix elements of $C$ can also be chosen randomly, as long as $C$ remains orthogonal.
I am hoping there to be some general form of the variance of the diagonal and off-diagonal elements of $\tilde{H}$, in terms of $\alpha$ and $\beta$. So far I have not been able to come up with it.
Any help is appreciated! Thanks!