## 4 Extensions of the Loewner Framework
Now, using the structure in (44) and in (46), it follows that:  
\[\mathbb{L}^{(i_{\bar{d}})} =-\mathcal{O}^{(i)}\,\mathbf{E}\,\mathcal{R}^{(\bar{j})} \tag{48}\] \[= \begin{bmatrix}\frac{\mathbf{H}_{0}(\mu^{(i)}_{1})-\mathbf{H}_{0} (\lambda^{(j)}_{1})}{\mu^{(1)}_{1}-\lambda^{(j)}_{1}}&\frac{\mathbf{H}^{ \mathbf{N}}_{1}(\mu^{(i)}_{1})-\mathbf{H}^{\mathbf{N}}_{1}(\lambda^{(j)}_{2} \lambda^{(j)}_{1})}{\mu^{(1)}_{1}-\lambda^{(j)}_{2}}\\ \frac{\mathbf{H}^{\mathbf{N}}(\mu^{(i)}_{1},\mu^{(i)}_{2})-\mathbf{H}^{ \mathbf{N}}_{1}(\mu^{(i)}_{1},\lambda^{(j)}_{1})}{\mu^{(2)}_{2}-\lambda^{(j)}_ {1}}&\frac{\mathbf{H}^{\mathbf{N}}_{2}(\mu^{(i)}_{1},\mu^{(i)}_{2},\lambda^{( j)}_{1})-\mathbf{H}^{\mathbf{N}}_{2}(\mu^{(i)}_{1},\lambda^{(j)}_{2} \lambda^{(j)}_{1})}{\mu^{(2)}_{2}-\lambda^{(j)}_{2}}\end{bmatrix} , \tag{49}\]  
and similarly for \(\mathbb{L}_{s}^{(i,j)}\). Hence, matrices \(\mathbb{L}\) and \(\mathbb{L}_{s}\) are indeed data matrices, since all their entries are samples of the system's transfer functions denoted with \(\mathbf{H}_{0}(s_{0}),\mathbf{H}^{\mathbf{N}}_{1}(s_{0},s_{1})\), and \(\mathbf{H}^{\mathbf{N},\mathbf{N}}_{2}(s_{0},s_{1},s_{2})\) (evaluated at particular grid of points). Next, introduce matrices:  
\[\mathbb{V}=\mathcal{O}\mathbf{B},\;\;\mathbb{W}=\mathbf{C}\mathcal{R},\;\; \mathbb{T}=\mathcal{O}\mathbf{N}\mathcal{R}, \tag{50}\]  
which can also be shown to be composed solely in terms of data, i.e., evaluations of the same transfer functions, introduced in (25). For matrix \(\mathbb{T}\), one can show that the \((i,j)\) block is explicitly written as follows:  
\[\mathbb{T}^{(i,j)}=\mathcal{O}^{(i)}\,\mathbf{N}\,\mathcal{R}^{(j)}= \begin{bmatrix} \mathbf{H}^{\mathbf{N}}_{1}(\mu^{(i)}_{1},\lambda^{(j)}_{1})&\mathbf{H}^{ \mathbf{N}}_{2}(\mu^{(i)}_{1},\lambda^{(j)}_{2},\lambda^{(j)}_{1})\\ \mathbf{H}^{\mathbf{N}}_{2}(\mu^{(i)}_{1},\mu^{(i)}_{2},\lambda^{(j)}_{1})& \mathbf{H}^{\mathbf{N}}_{3}(\mu^{(i)}_{1},\mu^{(i)}_{2},\lambda^{(j)}_{2}, \lambda^{(j)}_{1})\end{bmatrix} . \tag{51}\]  
As in the linear case, we focus on the case characterized by a redundant amount of data, which typically applies in practice. In such scenarios, the power of the Loewner framework relies on compressing and extracting the relevant information from the (possibly large) amount of data. Consequently, the singular value decomposition (SVD) is used as in the sequel. However, other factorization schemes, such as the QR factorization, or the CUR factorization [44], can indeed be used. Next, consider the (short) SVDs of the Loewner matrices (47). The matrices \(\mathbf{Y},\mathbf{X}\in\mathbb{C}^{k\times r}\) are obtained by selecting the first \(r\) columns of the matrices \(\mathbf{Y}_{1}\) and \(\mathbf{X}_{2}\). We define the following projection matrices (see also [46]):  
\[\mathbf{V}=\mathcal{R}\mathbf{X}\in\mathbb{C}^{n\times r},\quad\mathbf{W}= \mathcal{O}^{\star}\mathbf{Y}\in\mathbb{C}^{n\times r}. \tag{52}\]  
The Loewner ROM is given by the following matrices:  
\[\widehat{\mathbf{E}} =-\mathbf{Y}^{\star}\mathbb{L}\mathbf{X}=\mathbf{W}^{\star} \mathbf{E}\mathbf{V},\widehat{\mathbf{A}}=-\mathbf{Y}^{\star}\mathbb{L}_{s} \mathbf{X}^{\star}=\mathbf{W}^{\star}\mathbf{E}\mathbf{V}, \tag{53}\] \[\widehat{\mathbf{N}} =\mathbf{W}^{\star}\mathbf{N}\mathbf{V},\ \ \widehat{\mathbf{B}}= \mathbf{W}^{\star}\mathbf{B},\widehat{\mathbf{C}}=\mathbf{V}^{\star}\mathbf{C}.\]  
As in the linear case treated in [31], the projection matrices \(\mathbf{V},\mathbf{W}\), and the ROM have complex entries; however, one can enforce real-valued matrices of the ROM if the sets of left and of right interpolation points also contain the conjugate complex data as shown in [43] (one also needs to apply some specific transformations). As shown in (53), the Loewner ROM (53) can be computed using a Petrov-Galerkin projection (see also [1] for more details on this classical and widespread approach). However, this requires explicit access to the matrices of the original system.