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Say we have a matrix that has a singular value decomposition $M_1=U_1\Sigma_1V_1$, and we have another matrix $M_2$, which has the same number of rows. Can we say anything about the SVD of their concatenation, $M=[M_1 M_2]$ that reuses the SVD of $M_1$? In particular, if we know the decomposition of $M_1$ can we calculate the SVD of $M$ more quickly?

Purpose: I want to be able to speed up calculation of SVD if we've already computed SVD for a submatrix.

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    $\begingroup$ You might want to look into the effect of low rank perturbations on the SVD. This is especially useful when $M_1$ is $m \times n$, $M_2$ is $m \times p$, and $p \ll n$. $\endgroup$ Commented Dec 26, 2014 at 1:19
  • $\begingroup$ Leaving a comment to activate this question since it is 6 years old and no answers. $\endgroup$ Commented Nov 15, 2020 at 3:44

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