I come across this property while reading a paper and cannot find it anywhere: The property goes as follow: given a matrix $A \in R^{N\times n}$, denote $ U,V $ as the subspaces spanned by left and right singular vectors of $A$. Denote $P_U$ as the projection matrix on the subspace $U$. We have that$$A = P_U A$$ Furthermore, if $rank(A) = r$, denote $U_r$ as the subspace spanned by the first $r$ left singular vectors, and $P_{U_r}$ as the projection matrix to $U_r$, we have $$A=P_{U_r} A$$

I would really appreaciate if anyone can point out the name of this property and possibly show a sketch proof if it is simple enough.

Thanks so much.


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