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Bounding Frobenius Normnorm of Pseudopseudo-inverse

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

Bounding Frobenius Norm of Pseudo-inverse

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.

Bounding Frobenius norm of pseudo-inverse

$\DeclareMathOperator{\F}{\mathrm{F}}$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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Bounding Frobenius Norm of Pseudo-inverse

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.