Question

Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems​

Answer 1

Answer:

The problem you mentioned is a known problem of simultaneous similarity of pairs of matrices over a field and is called "wild" as it is basically reasonably considered hopeless in terms of finding a solution. It has been discovered a long time ago by Gelfand and Ponomaryov that the problem of simultaneous similarity of pairs of matrices contains the problem of simultaneous similarity of n-tuples of matrices for arbitrary given n. The problem has only been solved for specific (pretty low) dimensions. Overall, it is one of the typical problem types in modern algebra to determine whether a given classification problem (for instance, classification of group and algebra representations, linear groups, representations of posets, and more) to determine whether it's wild (i.e. contains the problem mentioned in your post) or tame (i.e. not wild). So, for instance, a group G is called wild over a commutative ring R if the problem of classification of all R-representations of G is wild. So, for instance, abelian (2,2)-group is tame over F2=Z/2Z, but (2,2,2) is wild over the same field.

Overall, finding tame classification problems and solving them is considered a productive thing to do.

You may find works of the following mathematicians helpful Dlab, Drozd, Kirichenko, Sergeichuk, Bondarenko. One of them can be found here: Complexity of Matrix Problems.