On varieties of commuting triples Original Research Article

The problem of approximating triples of commuting n×n matrices by triples of generic matrices is equivalent to that whether the variety C(3,n) of triples of commuting matrices is irreducible. It is known that the variety is irreducible for nless-than-or-equals, slant7 and reducible for ngreater-or-equal, slanted30. Using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we prove that C(3,8) is irreducible.