An 
Circulant matrices have the essential property that they’re diagonalized by the discrete Fourier remodel matrix
which satisfies 
Therefore circulant matrices are regular (
Observe that one specific eigenpair is speedy, since 
The factorization (1) permits a circulant linear system to be solved in 
A selected circulant matrix is the (up) shift matrix 
It’s not exhausting to see that
Since powers of
One essential use of circulant matrices is to behave as preconditioners for Toeplitz linear programs. A number of strategies have been proposed for setting up a circulant matrix from a Toeplitz matrix, together with copying the central diagonals and wrapping them round and discovering the closest circulant matrix to the Toeplitz matrix. See Chan and Ng (1996) or Chan and Jin (2017) for a abstract of labor on circulant preconditioners for Toeplitz programs.
An fascinating circulant matrix is anymatrix('core/circul_binom',n) within the Anymatrix assortment, which is the 
>> n = 6; C = anymatrix('core/circul_binom',n), svd(C)
C =
1 6 15 20 15 6
6 1 6 15 20 15
15 6 1 6 15 20
20 15 6 1 6 15
15 20 15 6 1 6
6 15 20 15 6 1
ans =
6.3000e+01
2.8000e+01
2.8000e+01
1.0000e+00
2.0244e-15
7.6607e-16
Notes
A basic reference on circulant matrices is Davis (1994).
References
- Raymond Chan and Michael Ng, Conjugate Gradient Strategies for Toeplitz Methods, SIAM Rev. 38(3), 427–482, 1996.
- Raymond Chan and Xiao-Qing Jin, An Introduction to Iterative Toeplitz Solvers, Society for Industrial and Utilized Arithmetic, Philadelphia, PA, USA, 2007.
- Philip Davis, Circulant Matrices, Second version, Chelsea, New York, 1994.
- Nicholas J. Higham and Mantas Mikaitis, Anymatrix: An Extendable MATLAB Matrix Assortment, Numer. Algorithms, 90:3, 1175-1196, 2021.
This text is a part of the “What Is” collection, out there from https://nhigham.com/class/what-is and in PDF kind from the GitHub repository https://github.com/higham/what-is.
