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$\DeclareMathOperator{\F}{\mathrm{F}}$Let $\mathbf{A}$ and $\mathbf{A}^\prime$ be two $m\times n$ matrix such that $\|\mathbf{A}-\mathbf{A}^\prime\|_{\F}\leq \delta$. Is there any bound for the difference of their pseudo-inverse? \begin{align} \|\mathbf{A}^{\mathrm{T}}(\mathbf{A}\mathbf{A}^{\mathrm{T}})^{-1}-\mathbf{A}'^{\mathrm{T}}(\mathbf{A}'\mathbf{A}'^{\mathrm{T}})^{-1}\|_{\F}\leq ?, \end{align} where $\|.\|_{\F}$ and $()^{\mathrm{T}}$ stand for Frobenius norm and transpose operator, respectively.

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  • $\begingroup$ Would a first-order bound in $\delta$ be sufficient for you? It is possible to get one by expanding $A' = A + E$ and using the identity $(A+E)^{-1} = A^{-1} - A^{-1}EA^{-1} + O(||E||^2)$. $\endgroup$ Commented Oct 13, 2020 at 19:45
  • $\begingroup$ $AA^t$ need not be invertible. $\endgroup$ Commented Oct 14, 2020 at 13:36
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    $\begingroup$ Assume that $n>m$ and $\mathbf{A}\mathbf{A}^{\mathrm{T}}$ is invertible. $\endgroup$
    – Math_Y
    Commented Oct 14, 2020 at 14:46

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