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 be a real matrix and . If is any characteristic root of , then

[4]

If is symmetric then and consequently the inequality implies that must be real.

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

Let and be the smallest and largest characteristic roots of , then

.
  1. ^ a b c 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.