# A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions
## 4 E-CATILED Technique
### Spectral Characterization of \(\beta\)-Matrices  
Given \(A\in\mathbb{C}^{n\times n}\), we below denote by \(A^{*}\) the transpose conjugate of \(A\). In this and the following subsection, for simplicity of notation, we consider the indices of the \(n\times n\)-matrices and \(n\)-vectors to vary between \(0\) and \(n-1\). We begin with presenting a class of simultaneously diagonalizable matrices recently proposed in [38]. Let \(n\) be a fixed positive integer, and \(Q_{n}=(q_{k,j})_{k,j}\in\mathbb{R}^{n\times n}\), where  
\[q_{k,j}=\left\{ \begin{array}{ll}\alpha_{j}\,\cos\Bigl{(}\frac{2\pi\,k\,j}{n }\Bigr{)}&\mbox{if $0\leq j\leq\lfloor n/2\rfloor$},\\ \\ \alpha_{j}\,\sin\Bigl{(}\frac{2\pi\,k\,(n-j)}{n}\Bigr{)}&\mbox{if $\lfloor n/2 \rfloor\leq j\leq n-1$},\end{array} \right. \tag{5}\]  
and  
\[\alpha_{j}=\left\{ \begin{array}{ll}\frac{1}{\sqrt{n}}=\widetilde{\pi}&\mbox{ if $j=0$, or $j=n/2$ if $n$ is even},\\ \\ \sqrt{\frac{2}{n}}=\widetilde{\pi}&\mbox{otherwise}.\end{array} \right.\]  
Set  
\[Q_{n}=\Bigl{(}\mathbf{q}^{(0)}\,\Big{|}\,\mathbf{q}^{(1)}\Big{|}\cdots\Big{|} \,\mathbf{q}^{(\lfloor\frac{n}{2}\rfloor)}\,\Big{|}\,\mathbf{q}^{(\lfloor \frac{n+1}{2}\rfloor)}\,\Big{|}\,\,\cdots\,\Big{|}\,\mathbf{q}^{(n-2)}\, \Big{|}\mathbf{q}^{(n-1)}\Bigr{)},\]  
with  
\[\mathbf{q}^{(0)}=\frac{1}{\sqrt{n}}\,\Bigl{(}1\,\,1\,\,\cdots\,\,1\Bigr{)}^{T }=\frac{1}{\sqrt{n}}\,\mathbf{u}^{(0)}, \tag{6}\]  
\[\mathbf{q}^{(j)} = \sqrt{\frac{2}{n}}\,\Bigl{(}1\,\,\,\cos\Bigl{(}\frac{2\pi j}{n} \Bigr{)}\,\cdots\,\cos\Bigl{(}\frac{2\pi j(n-1)}{n}\Bigr{)}\,\Bigr{)}^{T}= \sqrt{\frac{2}{n}}\,\mathbf{u}^{(j)},\] \[\mathbf{q}^{(n-j)} = \sqrt{\frac{2}{n}}\,\Bigl{(}0\,\,\sin\Bigl{(}\frac{2\pi j}{n} \Bigr{)}\,\cdots\,\sin\Bigl{(}\frac{2\pi j(n-1)}{n}\Bigr{)}\,\Bigr{)}^{T}= \sqrt{\frac{2}{n}}\,\mathbf{v}^{(j)}, \tag{7}\]  
\(j=1,2,\ldots,\lfloor\frac{n-1}{2}\rfloor\). If \(n\) is even, we have  
\[\mathbf{q}^{(n/2)}=\frac{1}{\sqrt{n}}\Bigl{(}1\,-1\,\,\,1\,-1\,\cdots\,-1 \Bigr{)}^{T}=\frac{1}{\sqrt{n}}\,\mathbf{u}^{(n/2)}. \tag{8}\]  
Note that \(Q_{n}\) is an orthonormal matrix (cf. [44]). Given a vector \(\mathbf{\lambda}\in\mathbb{C}^{n}\), \(\mathbf{\lambda}=(\lambda_{0}\,\lambda_{1}\cdots\lambda_{n-1})^{T}\), we define  
\[\text{diag}(\mathbf{\lambda})=\Lambda= \begin{pmatrix}\lambda_{0}&0&0&\ldots&0&0\\ 0&\lambda_{1}&0&\ldots&0&0\\ 0&0&\lambda_{2}&\ldots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\ldots&\lambda_{n-2}&0\\ 0&0&0&\ldots&0&\lambda_{n-1}\end{pmatrix} \in\mathbb{C}^{n\times n}.\]  
We recall that a vector \(\mathbf{\lambda}\in\mathbb{R}^{n}\), \(\mathbf{\lambda}=(\lambda_{0}\,\lambda_{1}\cdots\lambda_{n-1})^{T}\) is said to be _symmetric_ if \(\lambda_{j}=\lambda_{n-j}\) for \(j=0\), \(1,\ldots,\lfloor n/2\rfloor\); otherwise, \(\mathbf{\lambda}\) is said to be _asymmetric_ if \(\lambda_{j}=-\lambda_{n-j}\) for \(j=0\), \(1,\ldots,\lfloor n/2\rfloor\). Let \(\mathcal{G}_{n}\) be the space of the following simultaneously diagonalizable matrices  
\[\mathcal{G}_{n}=\text{sd}(Q_{n})=\{Q_{n}\Lambda Q_{n}^{T}:\Lambda=\text{diag}( \mathbf{\lambda}),\,\mathbf{\lambda}\in\mathbb{R}^{n}\}.\]  
The elements of such a class are called \(\gamma\)_-matrices_. Moreover, in [38], the following classes are presented  
\[\mathcal{C}_{n}=\{Q_{n}\Lambda Q_{n}^{T}:\Lambda=\text{diag}(\mathbf{\lambda}),\, \mathbf{\lambda}\in\mathbb{R}^{n},\,\mathbf{\lambda}\text{ is symmetric}\},\]  
\[\mathcal{B}_{n}=\{Q_{n}\Lambda Q_{n}^{T}:\Lambda=\text{diag}(\mathbf{\lambda}),\, \mathbf{\lambda}\in\mathbb{R}^{n},\,\mathbf{\lambda}\text{ is asymmetric}\},\]  
It is possible to see that \(\mathcal{G}_{n}\) is a matrix algebra of dimension \(n\), \(C_{n}\) is a subalgebra of \(\mathcal{G}_{n}\) of dimension \(\lfloor\frac{n}{2}\rfloor+1\), and \(\mathcal{B}_{n}\) is a linear subspace of \(\mathcal{G}_{n}\) of dimension \(\lfloor\frac{n-1}{2}\rfloor\) (see [45]). The following results hold. **Proposition 1** ([45]).