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Distinctness of the Eigenvalues of a Quadratic form in a Multivariate Sample
Open accessAbstract
This paper shows that a quadratic form in a multivariate sample has a certain rank and its nonzero eigenvalues are distinct with probability one under the assumption that the matrix defining the quadratic form satisfies a certain rank condition and that the underlying distribution of the sample is absolutely continuous with respect to Lebesgue measure.
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The solutions to a (nontrivial) polynomial are Lebesgue measure zero over the space of the parameters of the polynomial.
“Lemma 2 (Okamoto, 1973) . The solutions to a (nontrivial) polynomial are Lebesgue measure zero over the space of the parameters of the polynomial.”
From Robust agents learn causal world models (2024) · cites this paper as Okamoto, 1973
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“This involves converting a given constraint into polynomial equations over θ M and applying the following Lemma, Lemma 2 (Okamoto, 1973) .”
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Metadata
- Authors
- Okamoto, M.
- Corresponding author
- Okamoto, Masashi
- ORCID
- Okamoto, Masashi
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1973-07-01
- Volume
- 1
- Issue
- 4
- Pages
- 763-765
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W1975455467
- Cited by
- 181
- References
- 0
- Citations by year
- 2026: 11 · 2025: 4 · 2024: 2 · 2023: 5 · 2022: 5 · 2021: 6 · 2020: 5 · 2019: 6 · 2018: 1 · 2017: 7 · 2016: 3 · 2015: 8 · 2014: 4 · 2013: 13 · 2012: 5
- FWCI
- 2.08
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Random Matrices and Applications
- Keywords
- distinct eigenvalues, quadratic forms, absolutely continuous distributions, random matrices
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