Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality

L. Yu. Kolotilina

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XI, Tome 229 (1995), pp. 153-158

Résumé

Let $A=(a_{ij})^n_{i,j=1}$ be a Hermitian matrix and let $\lambda_1\geqslant\lambda_2\geqslant\dots\geqslant\lambda_n$ denote its eigenvalues. If $\sum^k_{i=1}=\lambda_i\sum^k_{i=1}a_{ii}$, $k<n$, then $A$ is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function $f(t)$, and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles.