What are the eigenvalues and eigenvectors of this circulant tridiagonal matrix?

Asked Jan 28 '20 at 02:51

Active Aug 05 '22 at 10:27

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\begin{equation} \begin{pmatrix} \alpha & \beta & 0 & \dots & 0 & 0 & \beta \\ \beta & \alpha & \beta & \dots & 0 & 0 & 0 \\ 0 & \beta & \alpha & \dots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & \dots & \alpha & \beta & 0 \\ 0 & 0 & 0 & \dots & \beta & \alpha & \beta \\ \beta & 0 & 0 & \dots & 0 & \beta & \alpha \end{pmatrix} \end{equation}

What is the closed-form solution for this circulant tridiagonal matrix?

edited Jan 28 '20 at 17:47

asked Jan 28 '20 at 02:51

kinder chen

1

Neither tridiagonal nor Toeplitz. It's circulant. The eigenvalues are related to the Discrete Fourier Transform. – Rodrigo de Azevedo Jan 28 '20 at 08:44
Thx, can you give more details? – kinder chen Jan 28 '20 at 16:55
See https://en.wikipedia.org/wiki/Circulant_matrix#Eigenvectors_and_eigenvalues – lhf Jan 28 '20 at 16:58
https://ee.stanford.edu/~gray/toeplitz.html – Rodrigo de Azevedo Jan 28 '20 at 19:01
@Rodrigo de Azevedo, thx for your note, but I still don't know how to implement the circulant matrix with DFT. Can you give a detailed example? – kinder chen Feb 04 '20 at 17:33
Similar: https://math.stackexchange.com/questions/2477418/finding-eigenvalues-in-almost-tridiagonal-matrix – Hans Lundmark Apr 16 '21 at 08:42

What are the eigenvalues and eigenvectors of this circulant tridiagonal matrix?

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