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 $\alpha =\underset{1\le i,j\le n}{\max }\frac{1}{2}|a_{ij}-a_{ji}|$. If $\lambda$ is any characteristic root of $A$, then

$|\mathrm{Im}(\lambda )|\le \alpha \sqrt{\frac{n(n-1)}{2}}.$

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 $\frac{A+A^H}{2}$, then

$m\le \mathrm{Re}(\lambda )\le M$.

See also

• Gershgorin circle theorem