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arXiv · 2105.14805

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

Hariprasad M.·Murugesan Venkatapathi
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Abstract

We begin by showing that any $n \times n$ 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 $\mathcal{O}(n^2)$ arithmetic operations. The utility of the approximate similarity transformation in preconditioning linear solvers is also demonstrated.

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Hariprasad M., Murugesan Venkatapathi. 2021-05-31. Circulant decomposition of a matrix and the eigenvalues of Toeplitz type matrices. https://arxiv.org/abs/2105.14805

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