Bendixson's inequality

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 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.

Mathematically, the inequality 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

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

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