A simple proof of the generalized Schur inequality (Q1322865)

Let \(A = (a_{ij})\) denote an \(n \times n\) matrix with eigenvalues \(\lambda_ 1, \dots, \lambda_ n\). The author gives a nice new proof of the generalized Schur inequality \[ \sum^ n_{i=1} | \lambda_ i |^ p \leq \sum^ n_{i,j = 1} | a_{ij} |^ p, \quad 1 \leq p<2 \] and describes all the matrices that convert the generalized Schur inequality into equality. The proof is further simplified for the special case \(p=1\).