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Bendixson's inequality — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Bendixson%27s_inequality
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-15T14:12:49.982Z
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Bendixson's inequality — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Bendixson%27s_inequality (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_39ce3cedcd91 (Metraipolis archive dc_39ce3cedcd91, sha256 e5f188dd9f2e8bd63afa0bc34c72c1bf1d95a26f52d09b65b18102c44ba2c6d0)

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@misc{dc39ce3cedcd91,
  title = {Bendixson's inequality — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Bendixson%27s_inequality},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_39ce3cedcd91; sha256 e5f188dd9f2e8bd63afa0bc34c72c1bf1d95a26f52d09b65b18102c44ba2c6d0},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Bendixson%27s_inequality. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary and real parts of characteristic roots (eigenvalues) of real matrices. A special case of this inequality leads to the result that characteristic roots of a real symmetric matrix are always real. The inequality relating to the imaginary parts of characteristic roots of real matrices is stated as: Let A = ( a i j ) {\displaystyle A=\left(a_{ij}\right)} be a real n × n {\displaystyle n\times n} matrix and α = max 1 ≤ i , j ≤ n 1 2 | a i j − a j i | {\displaystyle \alpha =\max _{1\leq i,j\leq n}{\frac {1}{2}}\left|a_{ij}-a_{ji}\right|} . If λ {\displaystyle \lambda } is any characteristic root of A {\displaystyle A} , then | Im ( λ ) | ≤ α n ( n − 1 ) 2 . {\displaystyle \left|\operatorname {Im} (\lambda )\right|\leq \alpha {\sqrt {\frac {n(n-1)}{2}}}.\,{}} If A {\displaystyle A} is symmetric then α = 0 {\displaystyle \alpha =0} and consequently the inequality implies that λ {\displaystyle \lambda } must be real. The inequality relating to the real parts of characteristic roots of real matrices (Theorem II in ) is stated as: Let m {\displaystyle m} and M {\displaystyle M} be the smallest and largest characteristic roots of A + A H 2 {\displaystyle {\tfrac {A+A^{H}}{2}}} , then m ≤ Re ( λ ) ≤ M {\displaystyle m\leq \operatorname {Re} (\lambda )\leq M} .

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Linear algebra»).
Fonte: Wikipedia in inglese, voce «Bendixson's inequality» — https://en.wikipedia.org/wiki/Bendixson%27s_inequality
Wikidata Q60791069 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q60791069
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Bendixson%27s_inequality?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1361335830, consultata il 2026-10-15T13:52:06.041Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 79abbdb45df2c729ddc7993b373dc0799604bcd27484c44a7d2d6c26dcb64f5f