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.
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Similarly:
- $e^{-\mathbf A t} e^{\mathbf A t} = \mathbf I$
Hence the result by definition of inverse matrix.
$\blacksquare$