In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902.[1] The inequality puts limits on the imaginary parts of Characteristic roots (Eigen values) of real matrices. A special case of this inequality leads to the result that characteristic roots of a real matrix are always real.
Mathematically, the inequality is stated as:
Let \( {\displaystyle A=\left(a_{ij}\right)} \) be a real \( n\times n \) matrix and \( {\displaystyle \alpha =\max _{1\leq i,j\leq n}{\frac {1}{2}}\left|a_{ij}-a_{ji}\right|} \). If \( \lambda \)is any characteristic root of A, then
\( {\displaystyle \left|\operatorname {Im} (\lambda )\right|\leq \alpha {\sqrt {\frac {n(n-1)}{2}}}.\,{}} \) [2]
If A is symmetric then \( \alpha =0 \) and consequently the inequality implies that \( \lambda \) must be real.
References
An Introduction to Linear Algebra. p. 210. Retrieved 14 October 2018.
Iterative Solution Methods. p. 633. Retrieved 14 October 2018.
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