Ranks and determinants of the sum of matrices from unitary orbits

The unitary orbit [image omitted] of an n n complex matrix A is the set consisting of matrices unitarily similar to A. Given two n n complex matrices A and B, ranks and determinants of matrices of the form X + Y with [image omitted] are studied. In particular, a lower bound and the best upper bound of the set [image omitted] are determined. It is shown that [image omitted] has empty interior if and only if the set is a line segment or a point; the algebraic structure of matrix pairs (A, B) with such properties are described. Other properties of the sets R(A, B) and (A,B) are obtained. The results generalize those of other authors, and answer some open problems. Extensions of the results to the sum of three or more matrices from given unitary orbits are also considered.