The Additive Inverse Eigenvalue Problem for Lie Perturbations
Christopher I. Byrnes, Xiaochang Wang · SIAM Journal on Matrix Analysis and Applications · 1993
Motivated by several examples arising in linear systems theory, the problem considered here is the inverse eigenvalue problem for an arbitrary square matrix and for arbitrary additive perturbations belonging to a matrix Lie algebra. For an algebraically closed field with characteristic zero, the main theorem gives necessary and sufficient conditions for the positive solution of the corresponding additive inverse eigenvalue problem. There are, of course, several antecedents of this result in the literature involving special coordinate systems and special representations of particular matrix Lie algebras, most notable among these being the result due to S. Friedland on the inverse eigenvalue problem for diagonal perturbations.