I am trying to go through this process using the chain rule: $$\frac{\partial tr(\sqrt{SS^T})}{\partial S}=\frac{\partial tr(\sqrt{SS^T}) }{\partial \sqrt{SS^T} }\cdot\frac{\partial \sqrt{SS^T}}{\partial S},$$
I am assuming the result $\frac{\partial tr(\sqrt{SS^T})}{\partial S}$ should also be an $n$-dimensional square matrix. If I am understanding correctly, according to the chain rule, the first part results: $$\frac{\partial tr(\sqrt{SS^T}) }{\partial \sqrt{SS^T} }=\mathbb{I}_n\in\mathbb{R}^{n\times n},$$ and the second part results(according to https://math.stackexchange.com/q/1320527): $$\frac{\partial \sqrt{SS^T}}{\partial S} = (\sqrt{S}^T\oplus\sqrt{S})^{-1}\in \mathbb{R}^{n^2\times n^2},$$ and my question is how to understand/interpret the "$\cdot$" in chain rule(Since we cannot just simply multiple a $\mathbb{R}^{n\times n}$ matrix with a $\mathbb{R}^{n^2\times n^2}$ matrix)? Or say how to use chain rule to calculate the result for this derivative properly?
Thanks!