Eigenvalues of Circulant Matrices and Rank-One Perturbations: A Survey

Circulant matrices form an important class of structured matrices whose spectral properties can be described explicitly using the discrete Fourier transform. In this survey, we review the eigenvalue structure of circulant matrices with real entries and examine how rank-one perturbations affect their spectra. A simple illustrative example is included to demonstrate the stability and variation of eigenvalues under low-rank modifications.