Hariprasad, M and Venkatapathi, M (2024) Circulant decomposition of a matrix and the eigenvalues of Toeplitz type matrices. In: Applied Mathematics and Computation, 468 .
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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. © 2023 Elsevier Inc.
| Item Type: | Journal Article |
|---|---|
| Publication: | Applied Mathematics and Computation |
| Publisher: | Elsevier Inc. |
| Additional Information: | The copyright for this article belongs to author. |
| Keywords: | Eigenvalues and eigenfunctions; Linear transformations; Matrix algebra, Circulant matrix; Circulants; Eigen-value; matrix; N-matrix; Periodic entry; Preconditioners; Similarity transformation; Sparse approximations; Toeplitz, Fast Fourier transforms |
| Department/Centre: | Division of Interdisciplinary Sciences > Computational and Data Sciences |
| Date Deposited: | 01 Mar 2024 06:10 |
| Last Modified: | 01 Mar 2024 06:10 |
| URI: | https://eprints.iisc.ac.in/id/eprint/83872 |
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