Gradient of $\operatorname{tr} \left( \left( C C^{T} \right)^{-1} \right)$

Question

What is the following gradient?

$$\nabla_C \operatorname{tr} \left( \left( C C^{T} \right)^{-1} \right)$$

Thank you for your attention.

Answer 1

For convenience, define the variable $$\eqalign{ X &= CC^T \cr dX &= dC\,C^T + C\,dC^T \cr\cr }$$ Write the function in terms of $X$ and find its differential $$\eqalign{ f &= {\rm tr}(X^{-1}) \cr df &= -X^{-2T}:dX \cr &= -X^{-2T}:(dC\,C^T + C\,dC^T) \cr &= -(X^{-2T}C:dC + C^TX^{-2T}:dC^T) \cr &= -(X^{-2T}C + X^{-2}C):dC \cr }$$ where a colon denotes the Frobenius Inner Product.

Since $df=\big(\frac{\partial f}{\partial C}:dC\big),\,$ the gradient must be $$\eqalign{ \frac{\partial f}{\partial C} &= -(X^{-2T} + X^{-2})\,C \cr }$$