Circulant decomposition of a matrix and the eigenvalues of Toeplitz type matrices

05/31/2021

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by   Hariprasad M., et al.

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We begin by showing that a n*n matrix can be decomposed into a sum of 'n' circulant matrices with appropriate relaxations. We use Fast-Fourier-Transform (FFT) operations to perform a sparse similarity transformation representing only the dominant circulant components, to evaluate all eigenvalues of dense Toeplitz, block-Toeplitz and other periodic or quasi-periodic matrices, to a reasonable approximation in O(n^2) arithmetic operations. This sparse similarity transformation can be exploited for other evaluations as well.