Matrix Exponential of Sum of Commutative Matrices

Theorem

Let $\mathbf A$ and $\mathbf B$ be a square matrices of order $m$ for some $m \in \Z_{\ge 1}$.

Let $t \in \R$ be a real number.

Let $e^{\mathbf A t}$ denote the matrix exponential of $\mathbf A$.

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

Then:

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

Proof

Let:

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

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