On singular values and similarity classes of matrices (Q1117004)

Let \({\mathcal M}_ n\) be the set of all \(n\times n\) complex matrices, \(A\in {\mathcal M}_ n\), \(\lambda_ 1,\lambda_ 2,...,\lambda_ n\) the eigenvalues of A arranged so that \(| \lambda_ 1| \geq | \lambda_ 2| \geq...\geq | \lambda_ n|\), \(\sigma^ 2_ 1,\sigma^ 2_ 2,...,\sigma^ 2_ n\) the eigenvalues of \(A^*A\) \((\sigma_ 1\geq \sigma_ 2\geq...\geq \sigma_ n\geq 0)\). Let s: \({\mathcal M}_ n\to R^ n\) be the map defined by \(s(A)=(\sigma_ 1,\sigma_ 2,...,\sigma_ n).\) The objective of the reviewed paper is to find the image \(s(\Delta_ A)\) where \(\Delta_ A\) is the similarity class containing A. The authors solve the problem when A is of cyclic type (i.e., the characteristic and minimal polynomials of A coincide), when A is nilpotent and in some other cases. The complete solution is given for \(n\leq 4\). Some conjectures are stated for the general case. At the beginning of the paper the authors present a very interesting review of known results relevant to the problem.