matrix exponential

矩阵指数
#linear_algebra

Definition

\begin{aligned} e^{At} \triangleq I + At + A^2\frac{t^2}{2!} + ... = \sum_{i=0}^\infty A^i \frac{t^i}{i!} \\ \frac{de^{At}}{dt} = A [I + At + A^2 \frac{t^2}{2!} + ...] = Ae^{At} \end{aligned}

Properties

$e^{A0} = I + A0 + A^2 \frac{0^2}{2!} + ... = I$
$(e^{At})^{-1} = e^{-At}$
$\frac{d}{dt} e^{At} = A e^{At}$

Notes

If $A$ is $1 \times 1$ i.e. scalar, this becomes the power series,

e^{at} = \sum_{k=0}^\infty \frac{(at)^k}{k!} = 1 + at + a^2\frac{t^2}{2!} + ...

which matches a common definition of the exponential function

References