We generalize the Wedderburn rank reduction formula by replacing the inverse with the Moore--Penrose pseudoinverse. In particular, this allows one to remove the non--singularity of a certain matrix from assumptions. The results implies in a straightforward way Nystroem, CUR decompositions, meta-factorization, and a result of Ameli, Shadden. We investigate which properties of the matrix are inherited by the generalized Wedderburn reduction. Reductions leading to the best low-rank approximation are explicitly described in terms of singular vectors. We give a self--contained calculation of the range and the nullspace of the projection $A(BA)^+B$ and prove that any projection can be expressed in this way.

翻译：我们通过用Moore-Penrose伪逆替代逆矩阵，推广了Wedderburn秩约简公式。特别地，这使得我们能够从假设中移除特定矩阵的非奇异性要求。该结果可直接推导出Nystroem分解、CUR分解、元因子分解以及Ameli和Shadden的一个结论。我们研究了矩阵的哪些性质在广义Wedderburn约简中得以保持。基于奇异向量，我们明确描述了能够产生最佳低秩近似的约简方法。我们独立计算了投影$A(BA)^+B$的值域与零空间，并证明任何投影均可通过此形式表示。