## 5 Numerical Simulations
### Modeling Matrix  
Referring back to (2), for an array of \(N_{a}\) elementary antennas, the number of baselines, and therefore the number of complex visibilities, is \(N_{b}\equiv N_{a}(N_{a}-1)/2\). Assuming that each elementary antenna also provides a measure of the average brightness temperature of the scene, the total number of real valued measurements is \(2N_{b}+N_{a}\equiv N_{a}^{2}\). As a consequence, the dimensions of the modeling matrix \(\mathbf{G}\) are therefore \(N_{a}^{2}\times N^{2}\), where \(N^{2}\) is the number of pixels of a sampling grid in the synthesized field of view. To fix the ideas, for the cross-shaped array shown in Figure 4, these dimensions are 27,889 \(\times\) 65,536. As a consequence of the geometry of the array, out of these 27,889 measurements, only 13,861 \(\equiv N_{f}\) are non-redundant: they correspond to the Fourier frequencies of the so-called experimental frequency coverage. The singular values decomposition [34] of the \(N_{a}^{2}\times N^{2}\equiv 27,889\times\) 65,536 rectangular matrix \(\mathbf{G}\) is a factorization that involves the \(N_{a}^{2}\equiv 27,889\) singular values \(\sigma\) ordered in descending order and two \(N_{a}^{2}\times N_{a}^{2}\equiv 27,889\times\) 27,889 and \(N^{2}\times N^{2}\equiv\) 65,536 \(\times\) 65,536 unitary matrices \(\mathbf{U}\) and \(\mathbf{V}\) so that the factorization reads  
\[\mathbf{G}=\sum_{\ell=1}^{N_{a}^{2}}\sigma_{\ell}\mathbf{U}_{\ell}\mathbf{V}_ {\ell}^{\mathrm{T}}\,, \tag{6}\]  
where \(\mathbf{U}_{\ell}\), resp. \(\mathbf{V}_{\ell}\), is the \(\ell\)th column of \(\mathbf{U}\), resp. \(\mathbf{V}\). Shown in Figure 5 are the distributions of the singular values \(\sigma_{\ell}\) of the modeling matrix \(\mathbf{G}\) of the antenna array of Figure 4 when the shortest spacing \(d\) between the elementary antennas is set to \(0.69\lambda\) or to \(0.71\lambda\), two values on each side of the frontier \(d/\lambda=\sqrt{2}/2\). In every case, two groups of singular values separated by a well-determined gap are observed: the first group is composed of the \(N_{f}\) largest singular values, the second one of the remaining smallest ones. In numerical linear algebra, the singular value decomposition can be used to determine the effective rank of a matrix [35], as rounding errors may lead to small but non-zero singular values with a rank deficient matrix [36]. From the numerical point of view, singular values beyond a given threshold are assumed to be numerically equivalent to zero. This threshold is often set to the so-called \(\epsilon\)-machine, which is equal to \(2^{-52}\simeq 2.2\cdot 10^{-16}\) in MATLAB [37]. The truncated singular value decomposition [38] is no longer an exact decomposition of the original matrix \(\mathbf{G}\), but rather provides the optimal low-rank matrix \(\widetilde{\mathbf{G}}\) approximating it:  
\[\mathbf{G}\simeq\widetilde{\mathbf{G}}=\sum_{\ell=1}^{Nf}\sigma_{\ell}\mathbf{ U}_{\ell}\mathbf{V}_{\ell}^{\mathrm{T}}, \tag{7}\]  
Figure 5: Distributions of the singular values of the modeling matrix \(\mathbf{G}\) of the cross-shaped array shown in Figure 4 for two values of the shortest spacing: (**a**) \(d=0.69\lambda\) and (**b**) \(d=0.71\lambda\). In every case, two groups of singular values separated by a well-determined gap are observed: the first group (green) is composed of the 13861 largest singular values, the second one (red) of the remaining smallest ones that could be considered as being equal to zero.