# Differentiating a Vector and a Matrix w.r.t. a Vector [Matrix Calculus]

I am studying matrix calculus for linear regression and machine learning and I would like to know exactly if the following calculations are correct:

Let $$y=\sin(x+yz)$$ and $$r=\begin{bmatrix}x\\y\\z\end{bmatrix}$$

Then the following is the gradient of $$y$$ i.e., the derivative of $$y$$ with respect the vector $$r$$ in denominator layout: $$\frac{\partial y}{\partial r}=\frac{\partial\sin(x+yz)}{\partial r}=\begin{bmatrix}\frac{\partial \sin(x+yz)}{\partial x}\\\frac{\partial\sin(x+yz)}{\partial y}\\\frac{\partial \sin(x+yz}{\partial z}\end{bmatrix}=\begin{bmatrix} \cos(x+yz)\\z\cos(x+yz)\\y\cos(x+yz)\end{bmatrix}$$

In numerator layout it would be:

$$[\cos(x+yz), z\cos(x+yz), y\cos(x+yz)]$$

Now, let $$\mathbf{y}=\begin{bmatrix} e^{xyz}\\x^2z\\yx\end{bmatrix}$$

So in numerator layout: $$\frac{\partial\mathbf{y}}{\partial r}=\begin{bmatrix} \frac{\partial e^{xyz}}{\partial x} & \frac{\partial e^{xyz}}{\partial y} & \frac{\partial e^{xyz}}{\partial z}\\ \frac{\partial x^2z}{\partial x} & \frac{\partial x^2z}{\partial y} & \frac{\partial x^2z}{\partial z}\\ \frac{\partial yx}{\partial x} & \frac{\partial yx}{\partial y} & \frac{\partial yx}{\partial z}\end{bmatrix}\\ =\begin{bmatrix} yze^{xyz} & xze^{xyz} & xye^{xyz}\\2xz& 0 & x^2 \\y & x & 0\end{bmatrix}$$

In denominator layout it would be the transpose of the above matrix?

Now, let

$$Y=\begin{bmatrix} x^2yz & xy^2z \\xyz^2 & \ln(xyz) \end{bmatrix}=\begin{bmatrix} Y_{11} & Y_{12} \\Y_{21} & Y_{22} \end{bmatrix}$$

Then $$\frac{\partial Y}{\partial r}=\begin{bmatrix} \frac{\partial Y_{11}}{\partial x} & \frac{\partial Y_{12}}{\partial x} & \frac{\partial Y_{11}}{\partial y} & \frac{\partial Y_{12}}{\partial y} & \frac{\partial Y_{11}}{\partial z} & \frac{\partial Y_{12}}{\partial z} \\ \frac{\partial Y_{21}}{\partial x} & \frac{\partial Y_{22}}{\partial x} & \frac{\partial Y_{21}}{\partial y} & \frac{\partial Y_{22}}{\partial y} & \frac{\partial Y_{21}}{\partial z} & \frac{\partial Y_{22}}{\partial z}\end{bmatrix}$$

Would this be correct? I'm worrying I mixed up the shape and/or the positions of the members of the matrices $$\frac{\partial\mathbf{y}}{\partial r}$$ and $$\frac{\partial Y}{\partial r}$$.

• In this video (towards 6:30): youtube.com/watch?v=WrH-jpJIqFQ , he says that it's a kronecker product between $\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{ \partial }{\partial z}\right)$ and $Y$ Aug 29 at 12:15