Schur's inequalities bound sums involving eigenvalues of a matrix by sums involving its entries. Let 
be an 
matrix with complex
(or real) entries and eigenvalues 
, 
, ..., 
, then
 |
(1)
|
![sum_(i=1)^n|Re[lambda_i]|^2<=sum_(i,j=1)^n|(a_(ij)+a^__(ji))/2|^2](/images/equations/SchursInequalities/NumberedEquation2.svg) |
(2)
|
![sum_(i=1)^n|Im[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.
