How to solve a system of non-linear equations under matrix form with two unknown quantities (idealy scalars but surely matricial quantities given the problemetic) ?

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 will be able to help me

I have 2 covariance matrices known Cov_1 and Cov_2 that I want to cross-correlate.

1) For this, I have performed a diagonalisation of each Fisher matrix F_1 (FISH_eigen_sp in code) and F_2 (FISH_eigen_xc in code) associated of Covariance matrices Cov_1 and Cov_2.

So, I have 2 different linear combinations that are uncorraleted, i.e just related by eigen values (1/sigma_i^2) as respect of their combination.

Then, I get the diagonal matrices D_1 and D_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 (with D_1 diagonal) and P_2 matrices (with D_2 diagonal matrix).

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 :

1/sigma_final^2 = 1/sigma_sp^2 + 1/sigma_xc^2

because sigma_final 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

(I don't know yet if a and b should be scalar or matricial quantities).

1) P_1 represents the eigen vectors when diagonalizing F_1=FISH_sp : it is a passing matrix.

2) P_2 represents the eigen vectors when diagonalizing F_2=FISH_xc : it is a passing matrix.

3) eigen_sp and eigen_xc are the eigen values respectively of F_1 and F_2 matrices.

4) We have the diagonal matrix D=diag(eigen_{sp}+eigen_{xc}), i.e a diagonal matrix with the sum of `eigen_sp+eigen_xc` on each diagonal element.

PROBLEM:

Initialy, I want to build a combination of these eigen vectors under the form a P_1 + b P_2 with quantities a and b such that :

5) We respect the orthogonality of P =a P_1 + b P_2 and the condition P^T=P^{-1}, giving

5.1) a^2 Id + ab P_1 P_2^T + ab P_2 P_1^T + b^2 Id = 0

and for the second :

5) P=a P_1 + b P_2 and P^{-1} F_1 P + P^{-1} F_2 P = D, giving then :

F_1(a P_1 + b P_2) + F_2(a P_1 + b P_2) = P D

and finally :

5.2) a F_1 P_1 + b F_1 P_2 + a F_2 P_1 + b F_2 P_2 - (a P_1 + b P_2) D = 0

How to solve this kind of problem with the 2 equations 5.1) and 5.2) where I am looking for quantities a and b (I don't know yet if scalar would be solutions, otherwise these would be matricial solutions) ?

Sorry if there is no code for instant but I wanted to point the fact that I want to set correctly the problematic of this approach before trying to implement.