Matrix Exponential

linear-algebra

Definition

Matrix Exponential

A matrix exponential is a term where a matrix $A$ is raised to euler’s number $e$.

$$e^A=\sum _{k=0}^{\infty }\frac{A^k}{k!}=\underset{k\to \infty }{\lim }{\left(I+\frac{A}{k}\right)}^k$$

¹

It is derived using the Taylor series of $e^x$:

$$\begin{array}{rl}{e}^x& =\underset{k\to \infty }{\lim }{\left(1+\frac{x}{k}\right)}^k\\ & =x^0+x^1+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\cdots +\frac{1}{k!}{x}^k\end{array}$$

²

Footnotes

1. Matrix exponential - Wikipedia ↩

2. Getting Some Intuition for Matrix Exponentials – Emma Benjaminson – Data Scientist ↩