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linear algebra – Maximal rank resolution to $W^+Y = W^+WX$ Answer

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linear algebra – Maximal rank resolution to $W^+Y = W^+WX$

Let $X in mathbb{R}^{a occasions n}$ and $Y in mathbb{R}^{b occasions n}$. How can we discover a matrix $W in mathbb{R}^{b occasions a}$ of maximal rank, such that
$$W^+Y = W^+WX$$
(or alternatively, $WW^+Y = WX$) the place $W^+$ denotes the pseudoinverse of $W$? We concern in regards to the $W$ of maximal rank, as one may trivially take $W$ to breathe all zeros and the equation would maintain. From a geometrical perspective, we’re searching for a $W$ such that when $X$ is mapped to the column house of $W$, and $Y$ is mapped to the row house of $W$, these two photos are equal when one considers the canonical isomorphism between the row and column house of $W$. In illustration it is needful, we are able to occupy that $X$ and $Y$ have complete rank.

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