I read that a circulant matrix $C$ can be written as $F \phi F^{-1}$ where $\phi$ are $C$'s eigenvalues. Can someone give me more information about the $F$ matrix? Will it be the same for any circulant matrix $C$? Is there any condition when I can't split a real circulant matrix $C$ as $F \phi F^{-1}$?
I read somewhere that $F$ is a unitary discrete Fourier transform matrix, and looks like this $[\omega^{i \cdot j}]_{NxN}$. In some other place I saw $F$ defined as $\frac{1}{\sqrt{N}}[\omega^{i \cdot j}]_{NxN}$. Can someone explain this difference?
https://en.wikipedia.org/wiki/DFT_matrix https://en.wikipedia.org/wiki/Discrete_Fourier_transform#The_unitary_DFT