Young tableaux and stratification of space of complex square matrices

A stratification of the manifold of all square matrices is considered. One equivalence class consists of the matrices with the same sets of non-vanishing values $\mathrm{rank}(A-\lambda_i\mathrm I)^j$. The stratification is consistent with a fibration on the submanifolds of matrices similar to each other, i.e. with the adjoint orbits fibration. Internal structures of the matrices from one equivalence class are very similar, among other factors their (co)adjoint orbits are canonically birationally symplectomorphic. A Young tableaux technic developed in the article describes this stratification and the fibration of the strata on the (co)adjoint orbits.