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Schur's Inequalities

Let ${\hbox{\sf A}}=a_{ij}$ be an $n\times n$ Matrix with Complex (or Real) entries and Eigenvalues $\lambda_1$, $\lambda_2$, ..., $\lambda_n$, then

\begin{displaymath}
\sum_{i=1}^n \vert\lambda_i\vert^2\leq \sum_{i,j=1}^n \vert a_{ij}\vert^2
\end{displaymath}


\begin{displaymath}
\sum_{i=1}^n \vert\Re[\lambda_i]\vert^2\leq \sum_{i,j=1}^n \left\vert{a_{ij}+a_{ji}^*\over 2}\right\vert^2
\end{displaymath}


\begin{displaymath}
\sum_{i=1}^n \vert\Im[\lambda_i]\vert^2\leq \sum_{i,j=1}^n \left\vert{a_{ij}-a_{ji}^*\over 2}\right\vert^2.
\end{displaymath}


References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1120, 1979.




© 1996-9 Eric W. Weisstein
1999-05-26