On normal forms for the similarity classes of matrices and pairs of matrices

We answer two questions posed 1998 in the book ‘Arnolds problems’. First, over any field k there is a representative system for the similarity classes of nxn-matrices which is a finite disjoint union of affine subspaces. And second, for n>1 an analogous statement fails for pairs of nxn-matrices over any algebraically closed field of characteristic 0.