## 2 Some Remarks on Circulant and \(\omega\)-Circulant Matrices
\qquad\forall s,t=0, \ldots,n-1.\]  
_A compact representation for \(\mathbf{C}_{\omega}\) is given by_  
\[\mathbf{C}_{\omega}=\sum_{j=0}^{n-1}\Pi_{n,\omega}^{j}\alpha_{j},\]  
_where \(\Pi_{n,\omega}\) is the matrix_  
\[\Pi_{n,\omega}= \begin{bmatrix}0&0&0&\ldots&0&\omega\\ 1&0&0&\ldots&0&0\\ 0&1&0&\ldots&0&0\\ 0&0&1&\ldots&0&0\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ 0&0&0&\ldots&1&0\end{bmatrix} .\]  
Note that setting \(\omega=1\) leads back to Definition 1. An analogous spectral decomposition can be derived. **Proposition 2**. : \(\mathbf{C}_{\omega}\) _can be diagonalized as_  
\[\mathbf{C}_{\omega}=F_{n,\omega}L_{n,\omega}F_{n,\omega}^{-1},\]  
_where_  
\[L_{n,\omega}=\text{diag}_{s=0,\ldots,n-1}\left(\sum_{j=0}^{n-1}\omega^{j/n} \text{exp}\left(2\pi\text{i}\frac{sj}{n}\right)\alpha_{j}\right)\]  
_is the diagonal matrix containing the eigenvalues of \(\mathbf{C}_{\omega}\) and_  
\[F_{n,\omega}=D_{\omega}F_{n},\qquad D_{\omega}=\text{diag}_{s=0,\ldots,n-1} \left(\omega^{-\frac{s}{n}}\right).\]Proof. : Let us write \(\omega=\rho\exp(\mathrm{i}\psi)\) with \(\rho>0\) and consider \(\sqrt[n]{\omega}=\sqrt[n]{\rho}\exp\left(\mathrm{i}\frac{\psi}{n}\right)\). Applying (1), the matrix \(\Pi_{n,\omega}\) is diagonalized as  
\[\Pi_{n,\omega}=\sqrt[n]{\omega}D_{\omega}\Pi_{n}D_{\omega}^{-1}=\sqrt[n]{ \omega}D_{\omega}F_{n}\Omega_{n}F_{n}^{*}D_{\omega}^{-1}\]  
and the thesis follows. The above factorization is equivalent to  
\[\Pi_{n,\omega}=\sqrt[n]{|\omega|}D_{\omega}F_{n}\Omega_{n,\omega}F_{n}^{*}D_{ \omega}^{-1}\]  
with  
\[\Omega_{n,\omega}=\mathrm{diag}_{s=0,\ldots,n-1}\bigg{(}\exp\bigg{(}\frac{2 \pi s+\psi}{n}\mathrm{i}\bigg{)}\bigg{)},\]  
which, compared with (2), gives a deeper understanding of the role of \(\omega\). As in the circulant case, we stress that for any fixed \(p,q\in\mathbb{N}\) the matrix \(\mathbf{C}_{\omega}=\sum\limits_{j=-p}^{q}\Pi_{n,\omega}^{j}\alpha_{j}\) is \(\omega\)-circulant. **Remark 2**. : _With \(\omega=\rho\exp(\mathrm{i}\psi)\), the eigenvalues of \(\mathbf{C}_{\omega}\) can be expressed as_  
\[\Lambda(\mathbf{C}_{\omega})=\bigg{\{}f\bigg{(}\frac{2\pi s+\psi}{n}\bigg{)},s =0,\ldots,n-1\bigg{\}},\]  
_where the function \(f(\theta):=\sum\limits_{j=0}^{n-1}\alpha_{j}\rho^{\frac{j}{n}}\exp(\mathrm{i} j\theta)\) is called (spectral) symbol of \(\mathbf{C}_{\omega}\)._  
To conclude this section, we extend the above definitions and properties to the block case. **Definition 3**. : _Consider \(\mathcal{A}=[\mathcal{A}_{0},\mathcal{A}_{1},\ldots,\mathcal{A}_{n-1}]\) with \(\mathcal{A}_{j}\in\mathbb{R}^{d\times d}\). A square matrix \(\mathcal{C}_{\omega}\in\mathbb{R}^{d^{2}\times d^{2}}\) is called \(\omega\)-\((d\times d)\)-block circulant and it is denoted with \(\mathcal{C}_{\omega}=\text{circ}_{\omega}(\mathcal{A})\) if it holds_  
\[(\mathcal{C}_{\omega})_{s,t}= \begin{cases}\mathcal{A}_{s-t\pmod{n},}&\text{ if }s>t,\\ \omega\mathcal{A}_{s-t\pmod{n},}&\text{if }s\leq t,\end{cases} \qquad\forall s,t=0, \ldots,n-1.\]  
_A compact representation for \(\mathcal{C}_{\omega}\) is given by_  
\[\mathcal{C}_{\omega}=\sum\limits_{j=0}^{n-1}\Pi_{n,\omega}^{j}\otimes \mathcal{A}_{j}.\]  
For \(\omega=1\), Definition 3 describes the notion of \((d\times d)\)-block circulant matrices, while, as expected, for \(d=1\) it reduces to Definition 2. As in the case of \(d=1\), for any fixed \(p,q\in\mathbb{N}\) the matrix \(\mathcal{C}_{\omega}=\sum\limits_{j=-p}^{q}\Pi_{n,\omega}^{j}\otimes\mathcal{A }_{j}\) is \(\omega\)-\((d\times d)\)-block circulant. **Proposition 3**.