Ranks and determinants of the sum of matrices from unitary orbits

The unitary orbit U(A) of an n x n complex matrix A is the set consisting of matrices unitarily similar to A. Given two n x n complex matrices A and B, ranks and determinants of matrices of the form X + Y with (X, Y) is an element of U(A) x U(B) are studied. In particular, a lower bound and the best upper bound of the set R(A, B)= {rank(X + Y) : X is an element of U(A), Y is an element of U(B)} are determined. It is shown that Delta(A, B) = {det(X + Y) : X is an element of U(A), Y is an element of U(B)} 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 Delta(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.