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 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 (Theorem II in ) is stated as:
Let and be the smallest and largest characteristic roots of , then
.