I'm trying to find an expression for the matrix exponential of a Kronecker product of two matrices, $\hat{c}$ and $\hat{D}$. The matrix $\hat{c}$ is a small real and symmetric $2\times 2$ matrix:
$$ \hat{c} = \left( \begin{matrix} 0 & x \\ x & 0 \end{matrix}\right). $$
The matrix $\hat{D}$ is either $512\times 512$ or $1024\times 1024$ and dense. I'd like to calculate the following:
$$ \exp\left(- i \delta \hat{c}\otimes \hat{D} \right). $$
I am aware of the relationship between matrix exponentiation and the Kronecker sum:
$$ \exp\left(\hat{A}\oplus \hat{B}\right) = \exp\left(\hat{A}\right) \otimes \exp\left(\hat{B}\right). $$
Does anyone have any experience in approaching problems like this?