How to find the following derivative?
$\frac{d}{du} (u^TAu)^{-1} $
I know how to find the derivative without the inverse, that's just 2Au. But what about that?
2026-05-03 08:50:23.1777798223
Derivative of inverse of a matrix expression?
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Find the differential of the scalar, then of its reciprocal, and finally the gradient. $$\eqalign{ \alpha &= u^TAu \cr d\alpha &= (du^TAu + u^TA\,du) = du^T(A+A^T)u \cr d\alpha^{-1} &= -\frac{d\alpha}{\alpha^2} = du^T\,\bigg(\frac{(A+A^T)u}{-\alpha^2}\bigg) \cr \frac{\partial \alpha^{-1}}{\partial u} &= \frac{(A+A^T)u}{-\alpha^2} \cr }$$