Bendixson's inequality

In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902.{{Cite journal |last=Bendixson |first=Ivar |date=1902 |title=Sur les racines d'une équation fondamentale |journal=Acta Mathematica |volume=25 |pages=359–365 |doi=10.1007/bf02419030 |s2cid=121330188 |issn=0001-5962|doi-access=free }}{{cite book |title=An Introduction to Linear Algebra |page=210 |isbn=9780486166445 |url=https://books.google.com/books?id=TteOFYtbIVQC&q=Bendixson%27s+inequality&pg=PA436 |accessdate=14 October 2018|last1=Mirsky |first1=L. |date=3 December 2012 |publisher=Courier Corporation }} The inequality puts limits on the imaginary and real parts of characteristic roots (eigenvalues) of real matrices.{{Cite journal |last=Farnell |first=A. B. |date=1944 |title=Limits for the characteristic roots of a matrix |journal=Bulletin of the American Mathematical Society |volume=50 |issue=10 |pages=789–794 |doi=10.1090/s0002-9904-1944-08239-6 |issn=0273-0979|doi-access=free }} 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 $A\; =\; \backslash left\; (\; a\_\{ij\}\; \backslash right\; )$ be a real $n\; \backslash times\; n$ matrix and $\backslash alpha\; =\; \backslash max\_\{\{1\backslash leq\; i,j\; \backslash leq\; n\}\}\; \backslash frac\{1\}\{2\}\; \backslash left\; |\; a\_\{ij\}\; -\; a\_\{ji\}\; \backslash right\; |$. If $\backslash lambda$ is any characteristic root of $A$, then

: $\backslash left\; |\; \backslash operatorname\{Im\}\; (\backslash lambda)\; \backslash right\; |\; \backslash le\; \backslash alpha\; \backslash sqrt\{\backslash frac\{n(n-1)\}\; 2\; \}.\backslash ,\{\}${{cite book |title=Iterative Solution Methods |page=633 |isbn=9780521555692 |url=https://books.google.com/books?id=hNpJg_pUsOwC&q=Bendixson%27s+inequality&pg=PA633 |accessdate=14 October 2018|last1=Axelsson |first1=Owe |date=29 March 1996 |publisher=Cambridge University Press }}

If $A$ is symmetric then $\backslash alpha\; =\; 0$ and consequently the inequality implies that $\backslash lambda$ must be real.

The inequality relating to the real parts of characteristic roots of real matrices (Theorem II in ) is stated as:

Let $m$ and $M$

be the smallest and largest characteristic roots of $\backslash tfrac\{A+A^H\}\{2\}$, then

:$m\; \backslash leq\backslash operatorname\{Re\}(\backslash lambda)\; \backslash leq\; M$.