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Diagonalization of Cyclic Matrices

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A cyclic $ n\times n$-matrix $ A$ can be diagonalised by means of Fourier matrix

$\displaystyle W = (w^{jk})_{j,k=0,\ldots,n-1}, \quad w = \exp(2\pi\mathrm{i}/n)\, : $

$\displaystyle \frac{1}{n} \overline{W} \left(\begin{array}{cccc} a_0 & a_{n-1... ...\vdots & \ddots & \vdots \\ 0 & \cdots & \lambda_n \end{array}\right)\, , $

where

$\displaystyle \lambda_\ell = \sum_{k=0}^{n-1} a_k w^{-k'\ell},\quad \ell=0,\ldots,n-1 $

are the eigenvalues of $ A$.
(Authors: App/Burkhardt/Höllig)

Example:

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  automatically generated 3/16/2005