Spectrally arbitrary factorization: the nonderogatory case

Abstract

It is known that a nonsingular, nonscalar, n-by-n complex matrix A may be factored as A = BC, in which the spectra of B and C are arbitrary, subject to detBdetC = detA. We show further that B and C may be taken to be nonderogatory, even when the target spectra include repeated eigenvalues. This is a major step in a broader question of how arbitrary the Jordan forms of B and C may be, given their target spectra. In the process, a number of tools are developed, such as a special LU factorization under similarity, that may be of independent interest.