Circulant Matrices

A circulant matrix is a special type of matrix where each row is a right cyclic shift of the row above it. It is completely defined by its first row. Every 3×3 circulant matrix shares the same set of complex eigenvectors, which form the discrete Fourier basis.

Observe eigenvalues in the complex plane

Open Circulant Matrices. The module computes the matrix-vector product step-by-step and plots the selected eigenvalue.

1. Set the first row: c0 = 0, c1 = 1, c2 = 0. Set the input vector: x1 = 1, x2 = 2, x3 = 3.
2. Click Step three times. The resulting vector is [2, 3, 1]^T, which is a cyclic permutation of the input.
3. Observe the complex plane. With "Fourier mode" set to 0, the eigenvalue is exactly 1 + 0i.
4. Change "Fourier mode" to 1. The eigenvalue is now -0.5 + 0.866i, which corresponds to ω.
5. Change "Fourier mode" to 2. The eigenvalue becomes -0.5 - 0.866i, which is ω².
6. Click Reset, then change c to [2, 0, 0] and x to [1, -1, 2]. Click Finish. The output is [2, -2, 4]^T, and all eigenvalues are exactly 2 for all modes, because C = 2I.