This might be a silly question, but i have been really curious about the following:
Consider the following function seen thru a single variable, say $\alpha$:
\begin{equation} f(\alpha) = \mathbf{x}^{H}(\alpha) \mathbf{A}^{-1}(\alpha) \mathbf{x}(\alpha) \end{equation} where $\mathbf{x}(\alpha) \in \mathbb{C}^{N \times 1}$ and $\mathbf{A}(\alpha) \in \mathbb{C}^{N \times N}$.
I am interested in the $1^{st}$ and $2^{nd}$ order derivatives of $f(\alpha)$ at $\alpha_0$. However, $\mathbf{A}(\alpha_0)$ is singular, and is rank $N-1$.
I just want to check, even though it might be clear, that the $1^{st}$ and $2^{nd}$ order derivatives of $f(\alpha)$ at $\alpha_0$ do not exist. I even tried regularising by substituing $\mathbf{A}(\alpha)$ by $\mathbf{A}(\alpha) + \gamma \mathbf{I}$, where $\gamma$ goes to zero, but the results i get are not satisfying, i.e. regularising does not help.
I am actually interested in the ratio $\frac{f'(\alpha_0)}{f''(\alpha_0)}$. It turns out that regularising does not help.
P.S: I know that i could have a pseudo-inverse instead of the inverse. But i am interested in $f(\alpha)$ peaking up at value $\alpha_0$. If i put a pseudo-inverse, then the latter would not occur.