= Fréchet derivative of a negative matrix power
{title2=$D(A^{-m})[H]=-\sum_{j=1}^{m}A^{-j}HA^{-(m+1-j)}$}
For a positive integer $m$, the negative <matrix power> is differentiable on the open <general linear group>. The <Neumann series> gives $(A+H)^{-1}=A^{-1}-A^{-1}HA^{-1}+O(\|H\|^2)$ locally around any invertible matrix. Applying the <product rule> to the $m$ factors of this inverse gives the displayed <Fréchet derivative>. Each summand chooses one inverse factor to differentiate while keeping the others in order.
The factor order matters. Only when $A$ and $H$ commute does the sum simplify to $-mA^{-(m+1)}H$. This is a useful example of why scalar differentiation rules for powers cannot be transferred to matrices without accounting for noncommutativity.
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