Let 
be an 
matrix with complex (or
real) entries and eigenvalues 
, 
, ..., 
, then
 |
(1)
|
![sum_(i=1)^n|R[lambda_i]|^2<=sum_(i,j=1)^n|(a_(ij)+a^__(ji))/2|^2](/images/equations/SchursInequalities/NumberedEquation2.svg) |
(2)
|
![sum_(i=1)^n|I[lambda_i]|^2<=sum_(i,j=1)^n|(a_(ij)-a^__(ji))/2|^2,](/images/equations/SchursInequalities/NumberedEquation3.svg) |
(3)
|
where 
is the complex conjugate.
Schur's Inequalities
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References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1120, 2000.Referenced on Wolfram|Alpha
Schur's InequalitiesCite this as:
Weisstein, Eric W. "Schur's Inequalities." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchursInequalities.html