Английская Википедия:Bendixson's inequality
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 <math>A = \left ( a_{ij} \right )</math> be a real <math>n \times n</math> matrix and <math>\alpha = \max_Шаблон:1\leq i,j \leq n \frac{1}{2} \left | a_{ij} - a_{ji} \right |</math>. If <math>\lambda</math> is any characteristic root of <math>A</math>, then
- <math>\left | \operatorname{Im} (\lambda) \right | \le \alpha \sqrt{\frac{n(n-1)} 2 }.\,{} </math>[4]
If <math>A</math> is symmetric then <math>\alpha = 0</math> and consequently the inequality implies that <math>\lambda</math> must be real.
The inequality relating to the real parts of characteristic roots of real matrices (Theorem II in [1]) is stated as:
Let <math>m</math> and <math>M </math> be the smallest and largest characteristic roots of <math>\tfrac{A+A^H}{2}</math>, then
- <math>m \leq\operatorname{Re}(\lambda) \leq M</math>.
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