Derivative of matrix-valued function

Question

Let $X=ZZ^\top$ where $Z$ is a full row rank $m\times n$ matrix and let $Y$ be a nonsingular $n\times n$ matrix. Consider the matrix-valued function $$ f(X) = ZYZ^\top. $$

My question: Does there exists a closed form expression for the differential $\mathrm{d}f(X)$?

Remark. I believe that my question can be reduced to finding a closed form expression of the differential $\mathrm{d}Z$ w.r.t. $X$, but I don't know if this is actually possible.

Answer 1

You could use vectorization to attack the problem.

First $$\eqalign{ X &= ZZ^T \cr dX &= I\,dZ\,Z^T + Z\,dZ^T\,I \cr dx &= \big((Z\otimes I) + (I\otimes Z)K\big)\,dz = A\,dz\cr }$$ where $K$ is the kronecker commutation matrix such that $${\rm vec}(Z^T) = K\,{\rm vec}(Z)$$

Next $$\eqalign{ F &= ZYZ^T \cr dF &= I\,dZ\,YZ^T + ZY\,dZ^T\,I \cr df &= \big((ZY^T\otimes I) + (I\otimes ZY)K\big)\,dz = B\,dz \cr }$$ Then assuming that $A^{-1}$ exists $$\eqalign{ df &= BA^{-1}\,dx \cr dF &= {\rm mat}(df) = {\rm mat}(BA^{-1}\,dx) \cr }$$ where mat() is the inverse of the vec() operator.

Answer 2

Assuming that the function is well defined , the simplest approach is to compute as follows: $$ df(X)(WW^\top)=\lim_{h\to0}\frac1h((Z+hW)Y(Z+hW)^\top-ZYZ^\top)=ZYW^\top+WYZ^\top. $$ The linear map $df(X)$ is being applyed to a vector $WW^\top$. Now you can use coordinates to write the derivative as a matrix.