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I have a square matrix $Y$ and I would like to find the solution $X$ for the following equation: $$X(X^TX)^{-1}=X(Y^TY)^{-1}$$

In this equation, we can suppose that $Y^TY$ is invertible. We could also rewrite the equation as: $$X\left(I-(Y^TY)^{-1}X^TX\right)=0$$ if this could be useful, at least there is no requirement that $X^TX$ is invertible in this form.

Is this a common equation or one where the solution is straightforward to write?

EDIT

As suggested in the comments, $X=Y$ will probably work!

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    $\begingroup$ How about using X=Y? $\endgroup$ Commented Sep 4, 2022 at 23:37
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    $\begingroup$ Probably worth a try! Indeed I am sometimes looking for a complex answer to a so simple question... $\endgroup$
    – PC1
    Commented Sep 5, 2022 at 0:03
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    $\begingroup$ There is more than one solution; for instance, $X = QY$ for every orthonormal $Q$. So it would be better to specify if you want a solution, or to describe all solutions. $\endgroup$ Commented Sep 5, 2022 at 11:43
  • $\begingroup$ I am looking for any solution as the important matrix is $X^TX$. So the ease of computation is more my concern here. Any solution is acceptable (so $X=Y$ is good). $\endgroup$
    – PC1
    Commented Sep 5, 2022 at 17:53

1 Answer 1

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Notice that $X(X^T X)^{-1}$ is the Moore-Penrose inverse of $X^T$. Using properties of pseudo-inverse we get

$(Y^TY)(X^T X)^\dagger = I$

For instance $X=Y$ or $X=\text{Cholesky}(Y^T Y)$

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