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.