Let $X(\Omega)$ be a positive-semi-definite matrix which is a function of a set of parameters $\Omega$. I am interested in both cases where the matrix is real, or is Hermitian.
What is the derivative of the square root of this matrix with respect to an individual parameter $\Omega_i$, i.e
$
{\partial_{\Omega_i}\sqrt{X(\Omega)}}
$
?
Can this derivative be reduced to a form in terms of ${\partial_{\Omega_i}X(\Omega)}$?
Best Answer
For typing convenience define the matrices $$ S=\sqrt{X},\quad \dot S=\frac{dS}{d\Omega_i},\quad \dot X=\frac{dX}{d\Omega_i},\quad M=\left(I\otimes S+S^T\otimes I\right)^{-\tt1} $$
Utilizing the vec operation one can proceed as follows. $$\eqalign{ SS &= X \\ S\dot S + \dot SS &= {\dot X} \\ (I\otimes S+S^T\otimes I)\operatorname{vec}(\dot S) &= \operatorname{vec}({\dot X}) \\ \operatorname{vec}(\dot S) &= M\operatorname{vec}({\dot X}) \\ \dot S &= \operatorname{reshape}\left(M\operatorname{vec}\big({\dot X}\big),\; {\rm size}\big(S\big)\right) \\ }$$ If $M$ does not exist, then there is no solution but it might be possible to use the Moore-Penrose pseudoinverse to obtain a least-squares solution.