Matrix Calculus (Denominator layout)

Table of contents

• 1. Calculus on Manifolds - Differentiation
  • examples
• 2. Differentiation Fomulas

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1. Calculus on Manifolds - Differentiation

$$\overrightarrow{\mathbf{f}}:{\mathbb{R}}^n\to {\mathbb{R}}^m$$ $$\overrightarrow{\mathbf{x}}\in {\mathbb{R}}^n:\text{domain}f$$ $$\frac{\partial \overrightarrow{\mathbf{f}}}{\partial \overrightarrow{\mathbf{x}}}=\left[\begin{array}{ccc}|& & |\\ \frac{\partial f_1}{\partial \overrightarrow{\mathbf{x}}}& \cdots & \frac{\partial f_m}{\partial \overrightarrow{\mathbf{x}}}\\ |& & |\end{array}\right]$$

examples

: always based on the denominator vectors (denominator-layout notation)

$$\frac{\partial f}{\partial \overrightarrow{\mathbf{x}}}=\left[\begin{array}{c}|\\ \frac{\partial f}{\partial \overrightarrow{\mathbf{x}}}\\ |\end{array}\right]\text{ : }\frac{\partial \text{scalar}}{\partial \text{vector}}$$ $$\frac{\partial \overrightarrow{\mathbf{f}}}{\partial x}=\left[\begin{array}{ccc}\frac{\partial f_1}{\partial x}& \cdots & \frac{\partial f_m}{\partial x}\end{array}\right]\text{ : }\frac{\partial \text{vector}}{\partial \text{scalar}}$$ $$\frac{\partial \overrightarrow{\mathbf{f}}}{\partial \overrightarrow{\mathbf{x}}}=\left[\begin{array}{cccc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_2}{\partial x_1}& \cdots & \frac{\partial f_m}{\partial x_1}\\ \frac{\partial f_1}{\partial x_2}& \frac{\partial f_2}{\partial x_2}& \cdots & \frac{\partial f_m}{\partial x_2}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial f_1}{\partial x_n}& \frac{\partial f_2}{\partial x_n}& \cdots & \frac{\partial f_m}{\partial x_n}\end{array}\right]\text{ : }\frac{\partial \text{vector}}{\partial \text{vector}}\text{ (Jacobian)}$$

2. Differentiation Fomulas

Link : Fomulas - Wiki

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