Bendixson's inequality

Last updated January 18, 2024

In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902.^[1]^[2] The inequality puts limits on the imaginary and real parts of characteristic roots (eigenvalues) of real matrices.^[3] 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 (Theorem I in ^[1]) is stated as:

Let $A=\left(a_{ij}\right)$ be a real $n\times n$ matrix and $\alpha =\max _{1\leq i,j\leq n}{\frac {1}{2}}\left|a_{ij}-a_{ji}\right|$ . If $\lambda$ is any characteristic root of $A$ , then

\left|\operatorname {Im} (\lambda )\right|\leq \alpha {\sqrt {\frac {n(n-1)}{2}}}.\,{}

^[4]

If $A$ is symmetric then $\alpha =0$ and consequently the inequality implies that $\lambda$ must be real.

The inequality relating to the real parts of characteristic roots of real matrices (Theorem II in ^[1]) is stated as:

Let $m$ and $M$ be the smallest and largest characteristic roots of ${\tfrac {A+A^{H}}{2}}$ , then

m\leq \operatorname {Re} (\lambda )\leq M

.

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References

1 2 3 Bendixson, Ivar (1902). "Sur les racines d'une équation fondamentale". Acta Mathematica. 25: 359–365. doi: 10.1007/bf02419030 . ISSN 0001-5962. S2CID 121330188.
↑ Mirsky, L. (3 December 2012). An Introduction to Linear Algebra. Courier Corporation. p. 210. ISBN 9780486166445 . Retrieved 14 October 2018.
↑ Farnell, A. B. (1944). "Limits for the characteristic roots of a matrix". Bulletin of the American Mathematical Society. 50 (10): 789–794. doi: 10.1090/s0002-9904-1944-08239-6 . ISSN 0273-0979.
↑ Axelsson, Owe (29 March 1996). Iterative Solution Methods. Cambridge University Press. p. 633. ISBN 9780521555692 . Retrieved 14 October 2018.

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[:0-1] 1 2 3 Bendixson, Ivar (1902). "Sur les racines d'une équation fondamentale". Acta Mathematica. 25: 359–365. doi: 10.1007/bf02419030 . ISSN 0001-5962. S2CID 121330188.

[2] Mirsky, L. (3 December 2012). An Introduction to Linear Algebra. Courier Corporation. p. 210. ISBN 9780486166445 . Retrieved 14 October 2018.

[3] Farnell, A. B. (1944). "Limits for the characteristic roots of a matrix". Bulletin of the American Mathematical Society. 50 (10): 789–794. doi: 10.1090/s0002-9904-1944-08239-6 . ISSN 0273-0979.

[4] Axelsson, Owe (29 March 1996). Iterative Solution Methods. Cambridge University Press. p. 633. ISBN 9780521555692 . Retrieved 14 October 2018.

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Bendixson's inequality

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References