Bendixson's inequality

In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902.¹² 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 is stated as:¹

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

${\displaystyle \left|\operatorname {Im} (\lambda )\right|\leq \alpha {\sqrt {\frac {n(n-1)}{2}}}.\,{}}$⁴

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

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

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

${\displaystyle m\leq \operatorname {Re} (\lambda )\leq M}$.