# A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions
## 4 E-CATILED Technique
### Spectral Characterization of \(\beta\)-Matrices
: _One has_  
\[\mathcal{G}_{n}=\mathcal{C}_{n}\oplus\mathcal{B}_{n},\]  
_where \(\oplus\) is the orthogonal sum and \(\langle\cdot,\cdot\rangle\) denotes the Frobenius product, defined by_  
\[\langle G_{1},G_{2}\rangle\ =\ tr(G_{1}^{T}G_{2}),\qquad G_{1},G_{2}\in \mathcal{G}_{n},\]  
_where \(tr(G)\) is the trace of the matrix \(G\)._  
We recall the definition of the classical Hartley matrix (see also [19] and the references therein). If \(n\) is odd, we have  
\[H_{n}=\frac{1}{\sqrt{n}}\Big{(}\mathbf{u}^{(0)}\ \ \mathbf{u}^{(1)}+\mathbf{v}^{(1 )}\ \ \ldots\ \ \mathbf{u}^{\left(\frac{n-1}{2}\right)}+\mathbf{v}^{\left(\frac{n-1}{2}\right)} \ \ \mathbf{u}^{\left(\frac{n-1}{2}\right)}-\mathbf{v}^{\left(\frac{n-1}{2}\right)}\ \ldots\ \ \mathbf{u}^{(1)}-\mathbf{v}^{(1)}\Big{)}.\]  
When \(n\) is even, we obtain  
\[H_{n}=\frac{1}{\sqrt{n}}\Big{(}\mathbf{u}^{(0)}\ \ \mathbf{u}^{(1)}+\mathbf{v}^{(1 )}\ldots\mathbf{u}^{\left(\frac{n}{2}-1\right)}+\mathbf{v}^{\left(\frac{n}{2} -1\right)}\ \mathbf{u}^{\left(\frac{n}{2}\right)}\ \mathbf{u}^{\left(\frac{n}{2}-1\right)}- \mathbf{v}^{\left(\frac{n}{2}-1\right)}\ldots\mathbf{u}^{(1)}-\mathbf{v}^{(1 )}\Big{)}.\]  
It is not difficult to see that  
\[H_{n}=Q_{n}\,Y_{n},\]where \(Y_{n}=(y_{k,j})_{k,j}\in\mathbb{R}^{n\times n}\), is a diagonal matrix with  
\[y_{k,j}=\left\{ \begin{array}{ll}1&\text{if $k=j=0$,}\\ \frac{1}{\sqrt{2}}&\text{if $k=j$ and $1\leq k\leq\frac{n-1}{2}$,}\\ \frac{1}{\sqrt{2}}&\text{if $k+j=n$ and $1\leq k\leq n-1$,}\\ -\frac{1}{\sqrt{2}}&\text{if $k=j$ and $\frac{n+1}{2}\leq k\leq n-1$,}\\ 0&\text{otherwise}\end{array} \right.\]  
if \(n\) is odd, and  
\[y_{k,j}=\left\{ \begin{array}{ll}1&\text{if $k=j=0$ or $k=j=\frac{n}{2}$,}\\ \frac{1}{\sqrt{2}}&\text{if $k=j$ and $1\leq k\leq\frac{n}{2}-1$,}\\ \frac{1}{\sqrt{2}}&\text{if $k+j=n$ and $1\leq k\leq n-1$,}\\ -\frac{1}{\sqrt{2}}&\text{if $k=j$ and $\frac{n}{2}+1\leq k\leq n-1$,}\\ 0&\text{otherwise}\end{array} \right.\]  
if \(n\) is even. Now, set  
\[\mathcal{H}_{n}=\text{sd}(H_{n})=\{H_{n}\Lambda H_{n}^{T}:\Lambda=\text{diag}( \Lambda),\,\lambda\in\mathbb{R}^{n}\}. \tag{9}\]  
It is not difficult to see that  
\[\mathcal{C}_{n} = \{Q_{n}\Lambda Q_{n}^{T}:\Lambda=\text{diag}(\Lambda),\,\lambda \in\mathbb{R}^{n},\,\lambda\text{ is symmetric}\}0\] \[= \{H_{n}\Lambda H_{n}^{T}:\Lambda=\text{diag}(\Lambda),\,\lambda \in\mathbb{R}^{n},\,\lambda\text{ is symmetric}\}.\]  
From (9) and (10), it follows that  
\[\mathcal{H}_{n}=\mathcal{C}_{n}\oplus\mathcal{F}_{n},\]  
where  
\[\mathcal{F}_{n}=\{H_{n}\Lambda H_{n}^{T}:\Lambda=\text{diag}(\Lambda),\, \lambda\in\mathbb{R}^{n},\,\lambda\text{ is asymmetric}\}.\]  
The _Fourier matrix_ is defined by \(F_{n}=(f_{k,l})_{k,l}\in\mathbb{C}^{n\times n}\), where  
\[f_{k,l}=\frac{1}{\sqrt{n}}\,\omega_{n}^{kl},\qquad k,l=0,1,\ldots,n-1,\]  
with \(\omega_{n}=e^{\frac{2\pi i}{n}}\). Let \(\mathcal{W}_{n}\) be the space of all real matrices _simultaneously diagonalizable_ by \(F_{n}\), that is,  
\[\mathcal{W}_{n}=\text{sd}(F_{n})=\{F_{n}\Lambda F_{n}^{*}\in\mathbb{R}^{n \times n}:\Lambda=\text{diag}(\Lambda),\,\lambda\in\mathbb{C}^{n}\}.\]  
It is not difficult to see that \(\mathcal{W}_{n}\) is a commutative matrix algebra. Moreover, we define the following class:  
\[\mathcal{A}_{n}=\{F_{n}\Lambda F_{n}^{*}:\Lambda=\text{diag}(\Lambda),\, \lambda\in(\text{i}\,\mathbb{R})^{n},\,\lambda\text{ is asymmetric}\}.\]  
Finally, we define the \(\beta\)_-matrices_ as the matrices belonging to the following set:  
\[\mathcal{V}_{n}=\mathcal{C}_{n}\oplus\mathcal{B}_{n}\oplus\mathcal{F}_{n} \oplus\mathcal{A}_{n}.\]