Hi,

I guess this could be a rather silly question, but I got a bit confused about the "numerator layout notation" and "denominator layout notation" when working with matrix differentiation:

It says that with the denominator layout notation, we interpret differentiation of ascalarwith respect to a vector as such: $\frac{\mathrm{d}L}{\mathrm{d}w_1}=[\frac{\mathrm{d}L}{\mathrm{d}w_{11}}\frac{\mathrm{d}L}{\mathrm{d}w_{12}} ... \frac{\mathrm{d}L}{\mathrm{d}w_{1n}}]^T$, $L$ a scalar and $w_1$ an $n$ x $1$ vector.

But what if we represent the scalar $L$ differently? e.g $L=w^Tx$, where $w$, $x \in \Bbb{R}^{n \times1}$.

Then we get $\frac{\mathrm{d}L}{\mathrm{d}w}=\frac{\mathrm{d}(w^Tx)}{\mathrm{d}w}=\frac{\mathrm{d}(x^Tw)}{\mathrm{d}w}=x^T$, which is a $1$ by $n$ vector. Doesn't this result disagree with the denominator layout notation? I read somewhere on the wiki that says one should stick to one type of notation, but if certain types of calculations favors one type of notation over the other, wouldn't that be problematic or confusing?