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Schur's Inequalities


Let A=a_(ij) be an n×n matrix with complex (or real) entries and eigenvalues lambda_1, lambda_2, ..., lambda_n, then

 sum_(i=1)^n|lambda_i|^2<=sum_(i,j=1)^n|a_(ij)|^2
(1)
 sum_(i=1)^n|R[lambda_i]|^2<=sum_(i,j=1)^n|(a_(ij)+a^__(ji))/2|^2
(2)
 sum_(i=1)^n|I[lambda_i]|^2<=sum_(i,j=1)^n|(a_(ij)-a^__(ji))/2|^2,
(3)

where z^_ is the complex conjugate.


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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.

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Schur's Inequalities

Cite this as:

Weisstein, Eric W. "Schur's Inequalities." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchursInequalities.html

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