$\DeclareMathOperator{\F}{\mathrm{F}}$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.