## 2 Spectral Analysis  
First, we report a few definitions regarding the spectral and singular value distribution, the notion of clustering and a few relevant relationships among the various concepts. Then, we present the main theoretical tool taken from [4] and we perform a spectral analysis of the various matrix-sequences. Numerical experiments and visualization results corroborating the analysis are presented in the last part of the section. Definition 1: Let \(\{A_{n}\}_{n}\) be a sequence of matrices, with \(A_{n}\) of size \(d_{n}\), and let \(\psi:D\subset\mathbb{R}^{t}\to\mathbb{C}^{r\times r}\) be a measurable function defined on a set \(D\) with \(0<\mu_{t}(D)<\infty\). * We say that \(\{A_{n}\}_{n}\) has an (asymptotic) singular value distribution described by \(\psi\), and we write \(\{A_{n}\}_{n}\sim_{\sigma}\psi\), if \[\lim_{n\to\infty}\frac{1}{d_{n}}\sum_{i=1}^{d_{n}}F(\sigma_{i}(A_{n}))=\frac{1 }{\mu_{t}(D)}\int_{D}\frac{\sum_{i=1}^{r}F(\sigma_{i}(\psi(\mathbf{x})))}{r} \mathrm{d}\mathbf{x},\qquad\forall\,F\in C_{c}(\mathbb{R}).\] (6)
* We say that \(\{A_{n}\}_{n}\) has an (asymptotic) spectral (or eigenvalue) distribution described by \(\psi\), and we write \(\{A_{n}\}_{n}\sim_{\lambda}\psi\), if \[\lim_{n\to\infty}\frac{1}{d_{n}}\sum_{i=1}^{d_{n}}F(\lambda_{i}(A_{n}))=\frac{ 1}{\mu_{t}(D)}\int_{D}\frac{\sum_{i=1}^{r}F(\lambda_{i}(\psi(\mathbf{x})))}{r }\mathrm{d}\mathbf{x},\qquad\forall\,F\in C_{c}(\mathbb{C}).\] (7)  
If \(A\in\mathbb{C}^{m\times m}\), then the singular values and the eigenvalues of \(A\) are denoted by \(\sigma_{1}(A),\ldots,\sigma_{m}(A)\) and \(\lambda_{1}(A),\ldots,\lambda_{m}(A)\), respectively. Furthermore, if \(A\in\mathbb{C}^{m\times m}\) and \(1\leq p\leq\infty\), then \(\|A\|_{p}\) denotes the Schatten \(p\)-norm of \(A\), i.e., the \(p\)-norm of the vector \((\sigma_{1}(A),\ldots,\sigma_{m}(A))\); see [5] for a comprehensive treatment of the subject. The Schatten \(\infty\)-norm \(\|A\|_{\infty}\) is the largest singular value of \(A\) and coincides with the spectral norm \(\|A\|\). The Schatten \(1\)-norm \(\|A\|_{1}\) is the sum of the singular values of \(A\) and coincides with the so-called trace-norm of \(A\), while the Schatten \(2\)-norm \(\|A\|_{2}\) coincides with the Frobenius norm of \(A\), which is of great popularity in the numerical analysis community because of its low computational complexity. At this point, we introduce the definition of clustering, which, as for the distribution notions, is a concept only of the asymptotic type. For \(z\in\mathbb{C}\) and \(e>0\), let \(B(z,\epsilon)\) be the disk with center \(z\) and radius \(\epsilon\), \(B(z,\epsilon)\doteq\{w\in\mathbb{C}:\,|w-z|<\epsilon\}\). For \(S\subseteq\mathbb{C}\) and \(\epsilon>0\), we denote by \(B(S,\epsilon)\) the \(\epsilon\)-expansion of \(S\), defined as \(B(S,\epsilon)\doteq\bigcup_{z\in S}B(z,\epsilon)\). Definition 2: Let \(\{A_{n}\}_{n}\) be a sequence of matrices, with \(A_{n}\) of size \(d_{n}\) tending to infinity, and let \(S\subseteq\mathbb{C}\) be a nonempty closed subset of \(\mathbb{C}\). \(\{A_{n}\}_{n}\) is strongly clustered at \(S\) in the sense of the eigenvalues if, for each \(\epsilon>0\), the number of eigenvalues of \(A_{n}\) outside \(B(S,\epsilon)\) is bounded by a constant \(q_{\epsilon}\) independent of \(n\). In symbols,  
\[q_{\epsilon}(n,S)\doteq\#\{j\in\{1,\ldots,d_{n}\}:\lambda_{j}(A_{n})\
ot\in B (S,\epsilon)\}=O(1),\quad\text{as $n\to\infty$.}\]  
\(\{A_{n}\}_{n}\) is weakly clustered at \(S\) if, for each \(\epsilon>0\),  
\[q_{\epsilon}(n,S)=o(d_{n}),\quad\text{as $n\to\infty$.}\]  
If \(\{A_{n}\}_{n}\) is strongly or weakly clustered at \(S\) and \(S\) is not connected, then the connected components of \(S\) are called sub-clusters. Of special importance in the theory of preconditioning is the case of spectral single point clustering where \(S\) is made up by a unique complex number \(s\). The same notions hold for the singular values, where \(s\) is a nonnegative number and \(S\) is a nonempty closed subset of the nonnegative real numbers.