Schur’s inequality

Theorem (Schur’s inequality) Let A be a square n×n matrix with real (or possibly complex entries). If λ1,,λn are the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A, and D is the diagonal matrixMathworldPlanetmath D=diag(λ1,,λn), then


where F is the Frobenius matrix norm. Equality holds if and only if A is a normal matrixMathworldPlanetmath.


Title Schur’s inequality
Canonical name SchursInequality
Date of creation 2013-03-22 13:43:30
Last modified on 2013-03-22 13:43:30
Owner matte (1858)
Last modified by matte (1858)
Numerical id 14
Author matte (1858)
Entry type Theorem
Classification msc 26D15
Classification msc 15A42
Related topic TraceOfAMatrix
Related topic WielandtHoffmanTheorem
Related topic FrobeniusMatrixNorm