Bendixson's inequality

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Published on April 22, 2022 2:02:03 AM ● Video Link: https://www.youtube.com/watch?v=eNwrvjwqz-g

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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 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.
Mathematically, the inequality is stated as:
Let



A
=

(

a

i
j


)



{\displaystyle A=\left(a_{ij}\right)}
be a real



n
×
n


{\displaystyle n\times n}
matrix and



α
=

max

1
≤
i
,
j
≤
n




1
2



|


a

i
j


−

a

j
i



|



{\displaystyle \alpha =\max _{1\leq i,j\leq n}{\frac {1}{2}}\left|a_{ij}-a_{ji}\right|}
. If



λ


{\displaystyle \lambda }
is any characteristic root of



A


{\displaystyle A}
, then





|

Im
⁡
(
λ
)

|

≤
α




n
(
n
−
1
)

2



.






{\displaystyle \left|\operatorname {Im} (\lambda )\right|\leq \alpha {\sqrt {\frac {n(n-1)}{2}}}.\,{}}
If



A


{\displaystyle A}
is symmetric then



α
=
0


{\displaystyle \alpha =0}
and consequently the inequality implies that



λ


{\displaystyle \lambda }
must be real.

Source: https://en.wikipedia.org/wiki/Bendixson%27s_inequality
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