Bendixson's inequality
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 [math]\displaystyle{ A = \left ( a_{ij} \right ) }[/math] be a real [math]\displaystyle{ n \times n }[/math] matrix and [math]\displaystyle{ \alpha = \max_{{1\leq i,j \leq n}} \frac{1}{2} \left | a_{ij} - a_{ji} \right | }[/math]. If [math]\displaystyle{ \lambda }[/math] is any characteristic root of [math]\displaystyle{ A }[/math], then
- [math]\displaystyle{ \left | \operatorname{Im} (\lambda) \right | \le \alpha \sqrt{\frac{n(n-1)} 2 }.\,{} }[/math][4]
If [math]\displaystyle{ A }[/math] is symmetric then [math]\displaystyle{ \alpha = 0 }[/math] and consequently the inequality implies that [math]\displaystyle{ \lambda }[/math] must be real.
The inequality relating to the real parts of characteristic roots of real matrices (Theorem II in [1]) is stated as:
Let [math]\displaystyle{ m }[/math] and [math]\displaystyle{ M }[/math] be the smallest and largest characteristic roots of [math]\displaystyle{ \tfrac{A+A^H}{2} }[/math], then
- [math]\displaystyle{ m \leq\operatorname{Re}(\lambda) \leq M }[/math].
See also
References
- ↑ 1.0 1.1 1.2 Bendixson, Ivar (1902). "Sur les racines d'une équation fondamentale". Acta Mathematica 25: 359–365. doi:10.1007/bf02419030. ISSN 0001-5962.
- ↑ Mirsky, L. (3 December 2012). An Introduction to Linear Algebra. Courier Corporation. p. 210. ISBN 9780486166445. https://books.google.com/books?id=TteOFYtbIVQC&q=Bendixson%27s+inequality&pg=PA436. 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. https://books.google.com/books?id=hNpJg_pUsOwC&q=Bendixson%27s+inequality&pg=PA633. Retrieved 14 October 2018.
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