In a scientific paper, I’ve seen the following
δK−1δp=−K−1δKδpK−1
where K is a n×n matrix that depends on p. In my calculations I would have done the following
δK−1δp=−K−2δKδp=−K−TK−1δKδp
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=KK−1 respecting the order
(I)′⏟=0=(KK−1)′=K′K−1+K(K−1)′.
Solving this equation with respect to (K−1)′ (again paying attention to the order (!)) will give
K(K−1)′=−K′K−1⇒(K−1)′=−K−1K′K−1.
Attribution
Source : Link , Question Author : Sara , Answer Author : A.Γ.
