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

[math]\displaystyle{ \left | \operatorname{Im} (\lambda) \right | \le \alpha \sqrt{\frac{n(n-1)} 2 }.\,{} }[/math]

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

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

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

[math]\displaystyle{ m \leq\operatorname{Re}(\lambda) \leq M }[/math].

See also

• Gershgorin circle theorem