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 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 is stated as:
Let be a real matrix and. If is any characteristic root of, then
If is symmetric then and consequently the inequality implies that must be real.
The inequality relating to the real parts of characteristic roots of real matrices is stated as:
Let and be the smallest and largest characteristic roots of, then