Matrix Exponential

June 15, 2021

What is Matrix Exponential $e^{M t}$ ?

Wikipedia does have an entry on the topic, which I confess is a bit above my head. Instead I would like to recommend 3Blue1Brown’s excellent video How (and why) to raise e to the power of a matrix.

The rest of this article is simply my quick summary of what I think Matrix Exponential is.

Matrix Exponential $e^{M t}$ is a notation, with a definition similar to MacLaurin series of Exponential function.

e^{M t} = n = 0 \sum \infty \frac{1}{n !} M^{n} t^{n} = 1 + \frac{1}{1 !} M^{1} t^{1} + \frac{1}{2 !} M^{2} t^{2} + \frac{1}{3 !} M^{3} t^{3} \dots

This formulation is invented for solving diffential equation of the following form:

\frac{d}{d t} v (t) = M v (t)

A solution of the above differential equation is:

v (t) = e^{M t} v_{0}