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 and real 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.

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 $$n \times n$$ matrix and $$\alpha = \max_ \frac{1}{2} \left | a_{ij} - a_{ji} \right |$$. If $$\lambda$$ is any characteristic root of $$A$$, then


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

If $$A$$ is symmetric then $$\alpha = 0$$ and consequently the inequality implies that $$\lambda$$ 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 $$\tfrac{A+A^H}{2}$$, then


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