On singular values and similarity classes of matrices

AbstractLet s be the map Mn → Rn, where Mn is the space of n × n complex matrices, which assigns to each matrix A the n-tuple of its singular values arranged in decreasing order. We are interested in describing the image s(Δ) of a similarity class of matrices Δ ⊂ Mn. The problem is completely solved when Δ consists of cyclic matrices and in a number of other cases. We tabulate descriptions of s(Δ) for all similarity classes Δ ⊂ Mn when 2⩽n⩽4