Derivative of the inverse of a matrix

In a scientific paper, I’ve seen the following

δK1δp=K1δKδpK1

where K is a n×n matrix that depends on p. In my calculations I would have done the following

δK1δp=K2δKδp=KTK1δKδp

Is my calculation wrong?

Note: I think K is symmetric.

Answer

The major trouble in matrix calculus is that the things are no longer commuting, but one tends to use formulae from the scalar function calculus like (x(t)1)=x(t)2x(t) replacing x with the matrix K. One has to be more careful here and pay attention to the order. The easiest way to get the derivative of the inverse is to derivate the identity I=KK1 respecting the order
(I)=0=(KK1)=KK1+K(K1).
Solving this equation with respect to (K1) (again paying attention to the order (!)) will give
K(K1)=KK1(K1)=K1KK1.

Attribution
Source : Link , Question Author : Sara , Answer Author : A.Γ.

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