The norm of the derivative of a matrix with respect to a vector

Question

Let $A(x) \in \mathbb{R}^{n\times n}$ be a matrix depending on a vector $x \in \mathbb{R}^{n}$. What can we say about the derivative of $f(x) = (\tau A(x)- I)^{-1}x_0$ with respect to $\tau > 0$ and $x_0 \in \mathbb{R}^{n}$? Are there any simplifications? For the context of the problem, I am looking for a $\tau = \tau(x,x_0)$ such that $\lVert D_xf(x) \rVert < 1$ holds for a skew-symmetric matrix $A(x)$.

Answer 1

$ \def\l{\tau} \def\G{{\Gamma}} \def\LR#1{\left(#1\right)} \def\frob#1{\left\| #1 \right\|_F} \def\qiq{\quad\implies\quad} \def\mt{\mapsto} \def\p{\partial} \def\grad#1#2{\frac{\p #1}{\p #2}} \def\c#1{\color{red}{#1}} \def\gradLR#1#2{\LR{\grad{#1}{#2}}} $For typing convenience, define the variables $$\eqalign{ b &= x_0 &\qiq db = dx_0 \\ C &= \LR{I-A\l}^{-1} &\qiq \c{dC = C\LR{A\:d\l}C} \\ }$$ Calculate the differential of the function $$\eqalign{ f &= -Cb \\ df &= -\c{dC}\,b \;-\; C\,db \\ &= -\c{C\LR{A\:d\l}C}b \;-\; C\,db \\ &= \LR{CAf}d\l \;-\; C\,db \\ }$$ from which the required gradients are easily identified $$\eqalign{ \grad f\l &= CAf, \qquad\quad \grad fb &= -C \\ }$$