i want to caculate inverse matrix, supposing initial matrix is in first order, and to expand inverse matrix in first order, but i dont know what $\frac{∂A^{-1}_{mn}}{∂a^{ij}}$ is。
i know that $\frac{dA^{-1}}{dt}=-A^{-1}\frac{dA}{dt}A^{-1}$
Is the following caculation correct?
Suppose $a^{ij}→a^{ij}+ɛ$，get $\frac{dA^{mn}}{dɛ}=1$ only if mn=ij
$$ \frac{∂A^{-1}_{mn}}{∂a^{ij}}=\frac{dA^{-1}_{mn}}{dɛ}=-A^{-1}_{mx}\frac{dA^{xy}}{dɛ}A^{-1}_{yn}=-A^{-1}_{mi}A^{-1}_{jn} $$
Exzample:
i'm studying general relativity, the easiest linear solution ( slightly curved spacetimes ) and Kerr metric in limit.
just write a metric like
$$ g_{μν}= \begin{pmatrix} 1+ɛ & & & j \\ & -1+ɛ & & \\ & & -1+ɛ & \\ j & & & -1+ɛ \end{pmatrix} ≈ \begin{pmatrix} 1 & & & \\ & -1 & & \\ & & -1 & \\ & & & -1 \end{pmatrix} $$
How to caculate $g^{μν}$ in first order? Is it $$ g^{μν}= \begin{pmatrix} 1-ɛ & & & j \\ & -1-ɛ & & \\ & & -1-ɛ & \\ j & & & -1-ɛ \end{pmatrix} $$
It is shown here: Prove if $\lim_{k \rightarrow \infty} A_k = A$, then $\lim_{k \rightarrow \infty} A_k^{-1} = A^{-1}$ that if $A \in GL(n, \mathbb{R})$ and $H \in M(n, \mathbb{R})$ with $\Vert H \rVert < \frac{1}{\lVert A^{-1} \rVert}$, then $$(A + H)^{-1} = A^{-1} - A^{-1}HA^{-1} + O(\lVert H \rVert^2).$$ Here $\lVert \cdot \rVert$ denotes the operator norm.