matrix calculus: $\frac {\partial \vec{x}}{\partial \vec{x}^{T}}$

Question

I'm getting confused by notation conventions. In matrix calculus, it makes sense that: $$\frac {\partial \vec{x}}{\partial \vec{x}} = I$$ where I is the identity matrix. Is it true that: $$\frac {\partial \vec{x}}{\partial \vec{x}^{T}} = J$$ where J is the exchange matrix? (https://en.wikipedia.org/wiki/Exchange_matrix)

Or can this even be defined properly according to the numerator layout convention, since the numerator varies downward and the denominator varies across [like in the usual definition of the Jacobian matrix], but having a transpose in the denominator throws things off? (https://en.wikipedia.org/wiki/Matrix_calculus#Numerator-layout_notation)

Thanks.

Answer 1

I assume that $x$ is a vector. You consider the application $f:x^T\in M_{1n}\rightarrow x\in\mathbb{R}^n$. It's a linear application. Then, its derivative is itself: $Df_x:h^T\in M_{1n}\rightarrow h\in \mathbb{R}^n$. If we use the canonical bases of $M_{1n}$ and $\mathbb{R}^n$, then the matrix associated to $Df_x$ is $I_n$.