Matrix Calculus (Denominator layout)
Table of contents
1. Calculus on Manifolds - Differentiation
\[\overrightarrow{\mathbf{f}} : \mathbb{R}^n \rightarrow \mathbb{R}^m\] \[\overrightarrow{\mathbf{x}} \in \mathbb{R}^n : \text{domain}f\] \[\frac{\partial \overrightarrow{\mathbf{f}}}{\partial \overrightarrow{\mathbf{x}}} = \begin{bmatrix} \vert & & \vert \\ \frac{\partial f_1}{\partial \overrightarrow{\mathbf{x}}} & \cdots & \frac{\partial f_m}{\partial \overrightarrow{\mathbf{x}}}\\ \vert & & \vert \\ \end{bmatrix}\]examples
: always based on the denominator vectors (denominator-layout notation)
\[\frac{\partial f}{\partial \overrightarrow{\mathbf{x}}} = \begin{bmatrix} \vert \\ \frac{\partial f}{\partial \overrightarrow{\mathbf{x}}} \\ \vert \\ \end{bmatrix} \text{ : } \frac{\partial \text{scalar}}{\partial \text{vector}}\] \[\frac{\partial \overrightarrow{\mathbf{f}}}{\partial x} = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \cdots & \frac{\partial f_m}{\partial x} \end{bmatrix} \text{ : } \frac{\partial \text{vector}}{\partial \text{scalar}}\] \[\frac{\partial \overrightarrow{\mathbf{f}}}{\partial \overrightarrow{\mathbf{x}}} = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_2}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_1} \\ \frac{\partial f_1}{\partial x_2} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_2} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_1}{\partial x_n} & \frac{\partial f_2}{\partial x_n} & \cdots & \frac{\partial f_m}{\partial x_n} \\ \end{bmatrix} \text{ : } \frac{\partial \text{vector}}{\partial \text{vector}} \text{ (Jacobian)}\]2. Differentiation Fomulas
Link : Fomulas - Wiki