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 $\alpha =\max _{1\leq i,j\leq n}{\frac {1}{2}}\left|a_{ij}-a_{ji}\right|$. 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.

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

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

See also

• Gershgorin circle theorem