Let $A$ be a $n\times n$ matrix that is a real, symmetric, positive semidefinite and has rank $r$.

I read about the *rank reduced $QR$ decomposition* (here for example) as the QR variant where $A=QR$ with $Q\in\mathbb R^{n\times r}$ a matrix with orthogonal columns and $R\in\mathbb R^{r\times n}$ an upper triangular matrix. In the case of full rank $r=n$, then the decomposition is unique if the diagonal of $R$ is positive.

My question is: **are there any assumptions on $R$ that lead to the uniqueness of the decomposition when $r<n$?** For example, what if the diagonal of $R$ has to be positive and sorted?