I'm trying to analytically find the inverse of the covariance matrix generated by the exponential covariance function (also known as Ornstein-Uhlenbeck kernel) in $\mathbb{R}$, that is
$K_{ij} = k(x_i, x_j) = \exp(-\frac{|x_i - x_j|}{\theta})$
Where $x_i \in \mathbb{R}$ and that $K$ is an $n\times n$ matrix.
Here is what I've thought about doing:
Since $K$ is a covariance matrix, I know it is positive semidefinite and so is $K^{-1}$. We can do a singular value decomposition on $K = UDU^*$ and $K^{-1} = VCV^*$ where $U, V$ are unitary matrices and $D, C$ are diagonal matrices. But that I haven't been able to develop the idea further.
What would be right approach to get an analytical expression for $K^{-1}$?