Question

Every applied mathematician knows and uses the Singular Value Decomposition of a matrix, i.e., of products $U^T D V$, where $D$ is diagonal and $U, V$ are unitary. I am wondering if anything interesting can be said about products of the form $D U E$, in which $D, E$ are diagonal, and $U$ is unitary. What about their range and null space? And more generally what about the SVD of these matrices, in which we essentially swap the position of diagonal and unitary matrices? Any pointer to the literature is welcome. If $D, E$ are nonnegative and $U \ne I$, is it true that $tr(DUE)>tr(DE)$?