Firstly, I want to give you a maximum of informations and precisions about my issue. If I can’t manage to get the expected results, I will launch a bounty, maybe some experts or symply people who have been already faced to a similar problem would be able to help me.
1)
I have 2 covariance matrices known $Cov_1$ and $Cov_2$ that I want to crosscorrelate. (Covariance matrix is the inverse of Fisher matrix).
I describe my approach to crosscorrelate the 2 covariance matrices (the constraints are expected to be better than the constraints infered from a “simple sum” (elements by elements) of the 2 Fisher matrices).

For this, I have performed a diagonalisation of each Fisher matrix $F_1$ and $F_2$ associated of Covariance matrices $Cov_1$ and $Cov_2$.

So, I have 2 different linear combinations of random variablethat are uncorraleted, i.e just related by eigen values ($1/sigma_i^2$) as respect of their combination.
These eigen values of diagonalising are contained into diagonal matrices $D_1$ and $D_2$.
2) I can’t build a “global” Fisher matrix directly by summing the 2 diagonal matrices since the linear combination of random variables is different between the 2 Fisher matrices.
I have eigen vectors represented by $P_1$ and $P_2$ matrices.
That’s why I think that I could perform a “global” combination of eigen vectors where I can respect the MLE (Maximum Likelihood Estimator) as each eigen value :
$$dfrac{1}{sigma_{hat{tau}}^{2}}=dfrac{1}{sigma_1^2}+dfrac{1}{sigma_2^2}quad(1)$$
because $sigma_{hat{tau}}$ corresponds to the best estimator from MLE method.
So, I thought a convenient linear combination of each eigen vectors $P_1$ and $P_2$ that could allow to achieve it would be under a new matrix P whose each column represents a new eigein global vector like this :
$$P = aP_1 + bP_2$$
3) PROBLEM: : But there too, I can’t sum eigen values under the form $D_1 + D_2$ since the new matrix $P= a.P_1 + b.P_2$ can’t have in the same time the eigen values $D_1$ and also $D_2$ eigen_values, can it ?
I mean, I wonder how to build this new diagonal matrix $D’$ such that I could write :
$$P^{1} cdot F_{1} cdot P + P^{1} cdot F_{2} cdot P=D’$$
If $a$ and $b$ could be scalars, I could for example to start from taking the relation :
$$P^{1} cdot F_{1} cdot P = a^2*D_1quad(1)$$
and $$P^{1} cdot F_{2} cdot P = b^2*D_2quad(2)$$
with $(1)$ and $(2)$ making appear the relation : $$Var(aX+bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X,Y) = a^2 Var(X) + b^2 Var(Y)$$ since we are in a new basis $P$ that respect $(1)$ and $(2)$.
But the issue is that $a$ and $b$ seems to be matrices and not scalars, so I don’t know how to proceed to compute $D’$.
4) CONCLUSION :
Is this approach correct to build a new basis $P = a.P_1 + b.P_2$ and $D’ = a.a.D_1 + b.b.D_2$ assuming $a$ and $b$ are matrices ?
The key point is : if I can manage to build this new basis, I could return back to the starting space, the one of single parameters (no more combinations of them) by simply doing :
$$F_{text {cross}}=P . D’ cdot P^{1}$$ and estimate the constraints with covariance matrix : $C_{text{cross}}=F_{text {cross}}^{1}$.
If my approach seems to be correct, the most difficulty will be to determine $a$ and $b$ parameters (which is under matricial form, at least I think since with scalar form, there are too many equations compared to 2 unknown).
Sorry if there is no code for instant but I wanted to set correctly the problematic of this approach before trying to implement.
Hoping I have been clear enough.
Any help/suggestion/track/clue is welcome to solve this problem, this would be fine to tell it.