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

M. Hariprasad and Murugesan Venkatapathi

Applied Mathematics and Computation, 2024, vol. 468, issue C

Abstract: We begin by showing that any n×n matrix can be decomposed into a sum of n circulant matrices with periodic relaxations on the unit circle. This decomposition is orthogonal with respect to a Frobenius inner product, allowing recursive iterations for these circulant components. It is also shown that the dominance of a few circulant components in the matrix allows sparse similarity transformations using Fast-Fourier-transform (FFT) operations. This enables the evaluation of all eigenvalues of dense Toeplitz, block-Toeplitz, and other periodic or quasi-periodic matrices, to a reasonable approximation in O(n2) arithmetic operations. The utility of the approximate similarity transformation in preconditioning linear solvers is also demonstrated.

Keywords: Sparse approximation; Similarity transformation; Periodic entries; Fast Fourier transform; Preconditioners (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:468:y:2024:i:c:s0096300323006422

DOI: 10.1016/j.amc.2023.128473

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