Matlab : Study of the commutator with 2 Fisher matrices

问题

In Matlab, I have to study the eventual existence of common eigenvectors basis between 2 Fisher matrices FISH_sp and FISH_xc of size 7x7 and diagonalisable.

I get from my computation the following result:

>> x=null(FISH_sp*FISH_xc-FISH_xc*FISH_sp)

x =

   -0.0085
   -0.0048
   -0.2098
    0.9776
   -0.0089
   -0.0026
    0.0109

In this result, It appears that condition to get a common eigenvectors basis on commutator is true. But I need to further examine the mathematics. If one gets a single column vector, then nullspace of the commutator is 1-dimensional as far as Matlab can tell. With that result, one can think about how to verify that vector is indeed an eigenvector of FISH_sp and FISH_xc down to a small tolerance.

But I don't know how to introduce this tolerance in a small Matlab script.

All I have done for instant is :

x=null(FISH_sp*FISH_xc-FISH_xc*FISH_sp)

1. How can I introduce tolerance in the checking of eigenvector x as being really an eigenvector given a tolerance tol.

2. And what about the eigenvalues ? : normally, they should not equal to D1 in [V1, D1] =eig(FISH_sp)and not equal to D2 in [V2, D2] =eig(FISH_xc) ? I said they shouldn't since we have to express them in a new and different basis of eigenvectors : then I call these 2 news diagonal matrices D1_new and D2_new. So, I could write :

   If I have a passing matrix of all the common eigen vectors basis called P, then one has :

    F = P (D1_new + D2_new) p^-1
   

   This endomorphism F is wanted with this expression (to respect the Maximum Likelihood Estimator = MLE).

3. the problem for instant is that I have only one eigen vector x and not the entire passing matrix P of new eigenvectors. How can I build this passing matrix P from only the single x values of common eigen vector mentioned above ?

来源：https://stackoverflow.com/questions/65587856/matlab-study-of-the-commutator-with-2-fisher-matrices