# Fisher's formalism : How to find a complementary matrix to respect the Maximum Likelihood Estimator (MLE)?

I make following a previous post : Bad attempt to do cross-correlations between 2 matrices

Indeed, I say "Bad attempt" since the beginning of this study, I did a major error.

By wanting to diagonalise the 2 matrices, I get the variances of each matrix with a linear combination of random variables.

But I forgot to think that I can't simply sum the 2 diagonal elements (with variances of first matrix and variances of second matrix) to construct a final diagonal matrix. This reasoning is wrong since for each of the 2 matrix to cross-correlate, I have not the same combination of random variables. So it doesn't make sense to add the eigen values of each matrix.

At this step, the key point is : we may say that the sum of each eigen vector of matrices correspond to eigen values $$\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}$$.

So the goal is to find an endomorphism that could allow us to add these eigen values, we could write :

1. Taking $$P_1$$ and $$P_2$$ the 2 passing matrices when I diagonalise initial $$M_1$$ and $$M_2$$ matrices that I want to cross-correlate :

What it would be good is to write, by adding a third matrix $$P_X$$ to determine that checks :

$$\text{var}(P_1+P_2+P_X) = \text{var}(P_1) + \text{var}(P_2)$$

but that would be also written : $$\text{var}(P_1+P_2+P_X) = \text{var}(P_1) + \text{var}(P_2) + \text{covar}(P_1+P_2, P_X)$$

1. Then, it follows the condition : $$\text{covar}(P_1+P_2, P_X)=0$$

This way, by holding this condition, I could take as eigen vectors equal to the sum $$P_1 + P_2 + P_X$$ and the sum of eigen values $$D_1+D_2$$ : this sum of eigen values would check :

$$\dfrac{1}{\sigma_{\hat{\tau}}^{2}}=\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}\quad(1)$$

corresponding to the sum of each diagonalised Fisher matrix $$D_1$$ and $$D_2$$.

This way, I could build a "super" passing matrix equal to $$P_1 + P_2 + P_X$$ that would respect

a combination of eigen vectors with the sum of eigen values $$\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}$$.

1. Finally, if my reasoning is correct, knowing $$P_1$$ and $$P_2$$, I have to find $$P_X$$ such that :

$$\text{covar}(P_1+P_2, P_X)=0$$

Could anyone tell me if I am on the right track or if all I have said is wrong, and if yes, are there any other alternatives ?

Any help/suggestion/clue is welcome.