Properties of Matrix Exponential

Theorem

In the following:

$\mathbf A$ and $\mathbf B$ are constant square matrices of order $m$ for some $m \in \Z_{\ge 1}$

$\mathbf P$ is a non-singular square matrix of order $m$

$t, s \in \R$ are arbitrary real numbers.

The matrix exponential $e^{\mathbf A t}$ has the following properties:

Derivative of Matrix Exponential

$\dfrac \d {\d t} e^{\mathbf A t} = \mathbf A e^{\mathbf A t}$

Determinant of Matrix Exponential is Non-Zero

$\det e^{\mathbf A t} \ne 0$

where $\det$ denotes the determinant.

Same-Matrix Product of Matrix Exponentials

$e^{\mathbf A t} e^{\mathbf A s} = e^{\mathbf A \paren {t + s} }$

Inverse of Matrix Exponential

$\paren {e^{\mathbf A t} }^{-1} = e^{-\mathbf A t}$

where $\paren {e^{\mathbf A t} }^{-1}$ denotes the inverse of $e^{\mathbf A t}$.

Product with Matrix Exponential of Commutative Matrices

Let $\mathbf A \mathbf B = \mathbf B \mathbf A$.

Then:

$e^{\mathbf A t} \mathbf B = \mathbf B e^{\mathbf A t}$

Matrix Exponential of Sum of Commutative Matrices

Let $\mathbf A \mathbf B = \mathbf B \mathbf A$.

Then:

$e^{\mathbf A t} e^{\mathbf B t} = e^{\paren {\mathbf A + \mathbf B} t}$

Series Expansion of Matrix Exponential

$\ds e^{\mathbf A t} = \sum_{n \mathop = 0}^\infty \frac {t^n} {n!} \mathbf A^n$

Decomposition of Matrix Exponential

$e^{\mathbf P \mathbf A \mathbf P^{-1} } = \mathbf P e^{\mathbf A} \mathbf P^{-1}$