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

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  • Hariprasad, M.
  • Venkatapathi, Murugesan

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.

Suggested Citation

  • Hariprasad, M. & Venkatapathi, Murugesan, 2024. "Circulant decomposition of a matrix and the eigenvalues of Toeplitz type matrices," Applied Mathematics and Computation, Elsevier, vol. 468(C).
    • Handle: RePEc:eee:apmaco:v:468:y:2024:i:c:s0096300323006422
    DOI: 10.1016/j.amc.2023.128473
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    1. M․, Hariprasad & Venkatapathi, Murugesan, 2021. "Semi-analytical solutions for eigenvalue problems of chains and periodic graphs," Applied Mathematics and Computation, Elsevier, vol. 411(C).
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