“On Robust Matrix Completion with Prescribed Eigenvalues”

Matrix completion with prescribed eigenvalues is a special kind of inverse eigenvalue problems. Thus far, only a handful of specific cases concerning its existence and construction have been studied in the literature. The general problem where the prescribed entries are at arbitrary locations with arbitrary cardinalities proves to be challenging both theoretically and computationally. This paper investigates some continuation techniques by recasting the completion problem as an optimization of the distance between the isospectral matrices with the prescribed eigenvalues and the affine matrices with the prescribed entries. The approach not only offers an avenue to solving the completion problem in its most general setting but also makes it possible to seek a robust solution that is least sensitive to perturbation.