In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902.
[1][2] 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 (Theorem I in [1]) is stated as: Let
{\displaystyle A=\left(a_{ij}\right)}
{\displaystyle \alpha =\max _{1\leq i,j\leq n}{\frac {1}{2}}\left|a_{ij}-a_{ji}\right|}
is any characteristic root of
and consequently the inequality implies that
The inequality relating to the real parts of characteristic roots of real matrices (Theorem II in [1]) is stated as: Let
be the smallest and largest characteristic roots of