The Spectral Density for the Matérn Covariance Function

Question

In Paper1, we're working with a linear functional approximation to a Gaussian Process, shown below.

In equation (8) of this paper, we have

$$V=\mathrm{Diag}(S^{-1}(\sqrt{\lambda_j}))$$ (there's already a typo: a missing square root)

where $V^{-1}$ is the column variance of the matrix-normal distribution, and $S$ is the spectral density of the covariance function we've decided to use for the Gaussian Process. The $\lambda_j$'s are the eigenvalues at each we must evaluate the spectral density.

Is this $V$ matrix well defined (typo free, except for the square root)?

I ask this because the authors state that eq. 8 is a generalization to the multidimensional output case of the prior defined in eq. 3:

$$f(x)\approx \sum^m_{j=1}f^{(j)} \phi^{(j)}(x), \text{ with } f^{(j)}\sim N(0, S(\sqrt{\lambda_j}))$$

where the $\phi^{(j)}$ are the eigenfunctions corresponding to the eigenvalues above. This is the linear functional approximation.

However, unless for the Matérn covariance function has a spectral density such that $S^{-1}(\sqrt{\lambda_i})=(S(\sqrt{\lambda_i}))^{-1}$, I do not see how it's correct...

Addendum: The Matérn Class of functions is given by $$k_M(r)=\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\frac{r\sqrt{2\nu}}{l}\right)^{\nu} K_{\nu}\left(\frac{r\sqrt{2\nu}}{l}\right)$$ where $r=\Vert x-x'\Vert$. The function $K_\nu$ is the Modified Bessel of the 2nd kind. And its spectral density is given by: $$S(x):=\frac{2^D\pi^{\frac{D}{2}}\Gamma(\nu+D/2)(2\nu)^{\nu}}{\Gamma(\nu)l^{2\nu}}\left(\frac{2\nu}{l^2}+4\pi^2 x^2\right)^{-(\nu+D/2)}$$, where $D$ is the dimension of $x$.

P.S.: For the definition of Matérn Covariance function and its spectral density, see chapter 4 of this book

Paper1: "Computationally Efficient Bayesian Learning of Gaussian Process State Space Models", Andreas Svensson, Arno Solin, Simo Särkkä and Thomas B. Schön (2016).

Paper2: "Hilbert Space Methods for Reduced-Rank Gaussian Process Regression", Arno Solin, Simo Särkkä, (2014)