Inverse of Matrix Exponential

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Theorem

Let $\mathbf A$ be a square matrix.

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

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

Then:

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

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


Proof

\(\ds e^{\mathbf A t} e^{-\mathbf A t}\) \(=\) \(\ds e^{\mathbf A \paren {t - t} }\) Same-Matrix Product of Matrix Exponentials
\(\ds \) \(=\) \(\ds e^{\mathbf 0}\) Definition of Matrix Scalar Product: $\mathbf A 0 = \mathbf 0$
\(\ds \) \(=\) \(\ds \mathbf I\) Matrix Exponential of Zero Matrix

where:

$\mathbf 0$ denotes the zero matrix of the appropriate order
$\mathbf I$ denotes the identity matrix of the appropriate order.



Similarly:

$e^{-\mathbf A t} e^{\mathbf A t} = \mathbf I$

Hence the result by definition of inverse matrix.

$\blacksquare$