Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation} be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ distinct points $x_1,\ldots, x_N \in \mathbb{R}$, the corresponding kernel matrix $\mathbf{K}$, with entries \begin{equation} K_{ij} = k({x}_i,{x}_j). \end{equation} In Carl Edward Rasmussen's book (http://www.gaussianprocess.org/gpml/chapters/RW.pdf, page 113), it is stated that the complexity (penalty) $\log \vert \mathbf{K}\vert$ of a Gaussian process model with kernel matrix $\mathbf{K}$ decreases with the lengthscale, i.e., \begin{equation} \frac{d\log \vert \mathbf{K} \vert}{d\theta} \leq 0. \end{equation} Even though this seems to be common knowledge among people who employ Gaussian processes, I am struggling to prove it, and would like to know how to do so.