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I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$.

I am interested in evaluating the derivatives of the real and imaginary components of $\mathbf{Z}$ with respect to the real and imaginary components of $\mathbf{Y}$,

\begin{equation} \left[\begin{array}{c c } \frac{\partial \Re({Z_{ij}})}{\partial \Re(Y_{st})} & \frac{\partial \Re({Z_{ij}})}{\partial \Im(Y_{st})} \\[0.25cm] \frac{\partial \Im({Z_{ij}})}{\partial \Re(Y_{st})} & \frac{\partial \Im({Z_{ij}})}{\partial \Im(Y_{st})} \end{array}\right]. \end{equation}

From (Hjørungnes 2011) I am aware that the complex derivative of $\mathbf{Y^+}$ takes the form, \begin{equation} \frac{d}{dY_{st}} \mathbf{Y^+} = -\mathbf{Y^+} \left(\frac{d}{dY_{st}} \mathbf{Y}\right)\mathbf{Y^+} + \mathbf{Y^+} \mathbf{Y^{+H}} \left(\frac{d}{dY_{ij}} \mathbf{Y}\right)^\mathbf{H} (\mathbf{I}-\mathbf{Y}\mathbf{Y^+}) + (\mathbf{I}-\mathbf{Y^+}\mathbf{Y}) \left(\frac{d}{dY_{st}} \mathbf{Y}\right)^\mathbf{H}\mathbf{Y^{+H}} \mathbf{Y^+}. \end{equation}

My question is whether I can then apply the Cauchy-Reinmann equations to get the the derivatives with respect to the real and imagianry components?

\begin{equation} \left[\begin{array}{c c } \Re\left(\frac{\partial{Z_{ij}}}{\partial {Y_{st}} }\right) & -\Im\left(\frac{\partial{Z_{ij}}}{\partial {Y_{st}} }\right) \\[0.25cm] \Im\left(\frac{\partial{Z_{ij}}}{\partial {Y_{st}} }\right) & \Re\left(\frac{\partial{Z_{ij}}}{\partial {Y_{st}} }\right) \end{array}\right ] = \left[\begin{array}{c c } \frac{\partial \Re({Z_{ij}})}{\partial \Re(Y_{st})} & \frac{\partial \Re({Z_{ij}})}{\partial \Im(Y_{st})} \\[0.25cm] \frac{\partial \Im({Z_{ij}})}{\partial \Re(Y_{st})} & \frac{\partial \Im({Z_{ij}})}{\partial \Im(Y_{st})} \end{array}\right ]. \end{equation}

I am aware that their application requires the function in question to be analytic, but i do not know if this satisfied in the case of a complex pseudo inverse... If not, how else may i go about evaluating these derivatives?

Reference: @book{Hjrungnes:2011:CMD:2011870, author = {Hjrungnes, Are}, title = {Complex-Valued Matrix Derivatives: With Applications in Signal Processing and Communications}, year = {2011}, isbn = {0521192641, 9780521192644}, edition = {1st}, publisher = {Cambridge University Press}, address = {New York, NY, USA}, }

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  • $\begingroup$ Cauchy-Riemann requires that the function you are differentiating depends on $z=x+iy$ but not on the conjugate $z^\ast$; since the pseudo inverse corresponds to the limit $Y^+=\lim_{\epsilon\rightarrow 0}(Y^\ast Y+\epsilon I)^{-1})Y^\ast$ involving the conjugate transpose $Y^\ast$ of $Y$, it would seem this requirement is violated. $\endgroup$
    – Carlo Beenakker
    Commented Aug 7, 2018 at 9:04
  • $\begingroup$ If $\mathbf{Y^+}$ is complex differentiable (which i am under the impression it is based on [Hjørungnes 2011] ), does this not mean by definition that the function is analytic? Or am i missing some subtlety... $\endgroup$
    – Joshua Meggitt
    Commented Aug 7, 2018 at 14:41
  • $\begingroup$ As far as the function $\mathbf{Y^+}$ holomorphic and if in case the conjugate transpose $\mathbf{Y^*}$ of $\mathbf{Y}$ depends on Z, then their might be a possibility to calculate these derivatives. $\endgroup$
    – Amit Kumar Jaiswal
    Commented Aug 8, 2018 at 15:32

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