# $\frac{d\bf{Ax}}{d\bf{x}'}=A$ makes sense as Ax becomes a column vector given x is a column vector, but I see elsewhere $\frac{d\bf{Ax}}{d\bf{x}}=A'$

$$\frac{d\bf{Ax}}{d\bf{x}'} = \bf{A}$$ makes sense as $$\bf{Ax}$$ becomes a column vector given $$\bf{x}$$ is a column vector and $$\bf{x}'$$ is a row vector and $$\bf{A}$$ is a $$n\times n$$ matrix, but I see elsewhere that states $$\frac{d\bf{Ax}}{d\bf{x}} = \bf{A}'$$, which doesn't makes sense (to me!) because it takes a derivative of $$\bf{Ax}$$ by the column vector which should place each partial column-wise. It only makes sense if somehow $$Ax$$ is a row vector such as $$\bf{x'A'}$$.

What is the explanation of this result?

• There are a lot of conventions for whether the derivative of something w.r.t a row should be a row or a column. Have you checked the wiki article on matrix calculus? Apr 29, 2019 at 1:37
• I found things called "Denominator Layout Notation", and if you follow their convention, it makes sense. Apr 29, 2019 at 2:01

Index notation often brings clarity to this kind of problem. The only thing you need to know is $$\frac{\partial x_i}{\partial x_j} = \delta_{ij} = \begin{cases} 1 & i = j \\ 0 & i\ne j \end{cases}$$

For the current problem $$y=Ax$$ we have \eqalign{ y_i &= A_{ij}x_j \cr \frac{\partial y_i}{\partial x_k} &= A_{ij}\frac{\partial x_i}{\partial x_k} = A_{ij}\delta_{ik} = A_{ik} \cr \frac{\partial(Ax)}{\partial x} &= A \cr } where summation over a repeated index is implied (aka Einstein convention).

Differentiating $$w^T = x^TA^T\,$$ produces \eqalign{ w_i &= x_jA^T_{ji} = A_{ij}x_j =y_i \cr \frac{\partial w_i}{\partial x_k} &= \frac{\partial y_i}{\partial x_k} = A_{ik} \cr \frac{\partial(x^TA^T)}{\partial x} &= A \cr } You don't need to concern yourself with metric tensors or dual bases unless you're working with a non-Euclidean space (general relativity) or an oblique coordinate system (crystallography).

In standard cartesian coordinates you don't need to distinguish between $$y^T$$ and $$y$$. It just happens to be a limitation of matrix notation which creates this confusion.

• Nevertheless, transposes can be expressed in index notation, so you can actually translate back to matrices if you want. Also this helps one to give meaning to $\frac{\partial x^T}{\partial x}$ and other similar objects. Apr 30, 2019 at 2:47
• Can you make it clear which quantities are vectors? When you put subscript on variables, I am not sure if it's a component of vector indexed or an indexed column/row vector. May 1, 2019 at 1:29
• @Ikuyasu All of the indexed quantities are scalar elements of the corresponding matrix or vector. For example, $x_k$ is the $k^{th}$ component of the vector $x$, while $A_{mn}$ is the element located in the $m^{th}$ row and $n^{th}$ column of the matrix $A$.
– greg
May 1, 2019 at 2:24