Let $X \in \mathbb{R}^{a \times n}$ and $Y \in \mathbb{R}^{b \times n}$. How can we find a matrix $W \in \mathbb{R}^{b \times a}$ of maximal rank, such that $$W^+Y = W^+WX$$ (or alternatively, $WW^+Y = WX$) where $W^+$ denotes the pseudoinverse of $W$? We care about the $W$ of maximal rank, as one could trivially take $W$ to be all zeros and the equation would hold. From a geometric perspective, we're looking for a $W$ such that when $X$ is mapped to the column space of $W$, and $Y$ is mapped to the row space of $W$, these two images are equal when one considers the canonical isomorphism between the row and column space of $W$. In case it's necessary, we can assume that $X$ and $Y$ have full rank.


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