Any positive semi-definite matrix $A$ can be decomposed into $$ A = Q \Lambda Q^\dagger, $$ where $Q$ is a unitary matrix and $\Lambda$ is a diagonal matrix. Now I would assume that there is a partial decomposition as well, where $$ A = U X U^\dagger, $$ where $U$ is a different unitary matrix and the matrix $$ X = \begin{bmatrix} D & 0 \\ 0 & B \\ \end{bmatrix}, $$ consists of a diagonal block $D$ and a block $B$.
Does such a decomposition exist? If yes, does it have a particular name? Also, (if yes) what algorithm can be used to produce such a decomposition?