Bendixson's Inequality

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 $A = \left ( a_ \right )$ be a real $n \times n$ matrix and $\alpha = \max_ \frac \left , a_ - a_ \right ,$. If $\lambda$ is any characteristic root of $A$, then

: $\left , \operatorname (\lambda) \right , \le \alpha \sqrt.\,$

If $A$ is symmetric then $\alpha = 0$ and consequently the inequality implies that $\lambda$ 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 $\tfrac$, then

:$m \leq\operatorname(\lambda) \leq M$.

See also

* Gershgorin circle theorem

References

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Abstract algebra
Linear algebra
Matrix theory