Eigenvalue and eigenvector decomposition of the discrete Fourier transform
Abstract
The principal results of this paper are listed as follows. 1) The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are{1, -1,j, -j}. 2) An eigenvector basis is constructed for the DFT. 3) The multiplicities of the eigenvalues are summarized for an N×N transform as follows.
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The discrete fractional Fourier transform matrix defined by the spectral expansion is unitary because the eigenvalues of the DFT matrix have unit magnitude.
“This transform matrix is unitary since the eigenvalues of the DFT matrix have unit magnitude [38], [41].”
From The discrete fractional Fourier transform (2000) · cites this paper as [41]
The DFT matrix has only four distinct eigenvalues, so its eigenvalues are in general degenerate and the eigenvector set is not unique.
“Since the DFT matrix has only four distinct eigenvalues [41], the eigenvalues are in general degenerate so that the eigenvector set is not unique.”
From The discrete fractional Fourier transform (2000) · cites this paper as [41]
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- Authors
- McClellan, J., & Parks, T.
- Affiliations
- Rice University (US)
- ORCID
- McClellan, J.
- Venue
- IEEE Transactions on Audio and Electroacoustics
- Published
- 1972-03
- Volume
- 20
- Issue
- 1
- Pages
- 66-74
- ISSN
- 0018-9278 (ISSN-L)
- OpenAlex
- W2102337953
- Cited by
- 257
- References
- 4
- Citations by year
- 2026: 3 · 2025: 5 · 2024: 5 · 2023: 32 · 2022: 5 · 2021: 8 · 2020: 4 · 2019: 9 · 2018: 13 · 2017: 8 · 2016: 3 · 2015: 6 · 2014: 11 · 2013: 11 · 2012: 14
- FWCI
- 0.54
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Audio and Electroacoustics (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Keywords
- discrete Fourier transform, eigenvalue decomposition, eigenvalues, eigenvector basis, eigenvalue multiplicity, matrix eigenvalues
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