Bendixson's inequality explained

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) is stated as:

Let

A = \left ( a _{ij} \right ) be a real n x n matrix and \alpha = max _{{1\leq} } \frac \left | a_ - a_ \right |. If λ is any characteristic root of A , then\left | \operatorname{Im} ( λ ) \right | \le \alpha \sqrt{ (n(n -1 ))/(2) }.{} ^[4]

If

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

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

Let

m and M be the smallest and largest characteristic roots of \tfrac{A + A ^{H}{2}} , thenm \leq\operatorname{Re}( λ ) \leq M .

See also

• Gershgorin circle theorem

Notes and References

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