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19.2: The Circulant Matrix- Nature of its Eigenstates

Last updated

May 11, 2024
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- 19.1: The Model
- 19.3: Comparison with Raising Operators

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The matrix we’ve constructed above has a very special property: each row is identical to the preceding row with the elements moved over one place, that is, it has the form

$\begin{equation} \left(\begin{array}{llll} c_{0} & c_{1} & c_{2} & c_{3} \\ c_{3} & c_{0} & c_{1} & c_{2} \\ c_{2} & c_{3} & c_{0} & c_{1} \\ c_{1} & c_{2} & c_{3} & c_{0} \end{array}\right) \end{equation}$

Such matrices are called circulants, and their properties are well known. In particular, we’ll show that the eigenvectors have the form $\left(1, \omega_{j}, \omega_{j}^{2}, \omega_{j}^{3}, \ldots, \omega_{j}^{N-1}\right)^{T}$ where $\omega_{j}^{N}=1$ .

Unit circle with N-1 frequency points on the circumference each separated by an angle 2 pi / from the nextN — Figure $\PageIndex{1}$

Recall the roots of the equation $z^{N}=1$ are N points equally spaced around the unit circle,

$\begin{equation} e^{2 \pi i n / N}, n=0,1,2, \ldots N-1 \end{equation}$

The standard mathematical notation is to label these points $\begin{equation} 1, \omega_{1}, \omega_{2}, \omega_{3}, \ldots, \omega_{N-1} \end{equation}$ as shown in the figure, but notice that $\omega_{j}=\omega_{1}^{j}$

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