Let $A \in \mathbb{R}^{n \times n}$. In many places i see that $A$ can be represented as a projected unitary encoding, i.e. the matrix can be written as $A=PU\tilde{P}$, where $P,\tilde{P}$ are orthogonal projectors, and $U$ is a unitary matrix.
Why is this the case? I never saw such decomposition in any course of linear algebra. In fact, it is not any of the LU,QR,SVD, eigenvalue,Cholesky decomposition that I can read here on wikipedia.
What is the algorithm to come up with the decomposition?