Gradient flow methods for matrix completion with prescribed eigenvalues

Matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. The challenge of such a completion problem lies in the intertwining of the cardinality and the location of the prescribed entries so that the inverse problem is solvable. An intriguing question is whether matrices can have arbitrary entries at arbitrary locations with arbitrary eigenvalues and how to complete such a matrix. Constructive proofs exist to a certain point (and those proofs, such as the classical Schur-Horn theorem, are amazingly elegant enough in their own right) beyond which very few theories or numerical algorithms are available. In this paper the completion problem is recast as one of minimizing the distance between the isospectral matrices with the prescribed eigenvalues and the affined matrices with the prescribed entries. The gradient flow is proposed as a numerical means to tackle the construction. This approach is general enough that it can be used to explore the existence question when the prescribed entries are at arbitrary locations with arbitrary cardinalities.