Geodesic
Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XI, Tome 229 (1995), pp. 153-158 
L. Yu. Kolotilina. Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XI, Tome 229 (1995), pp. 153-158. http://geodesic.mathdoc.fr/item/ZNSL_1995_229_a4/
@article{ZNSL_1995_229_a4,
     author = {L. Yu. Kolotilina},
     title = {Interrelations between eigenvalues and diagonal entries of {Hermitian} matrices implying their block diagonality},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {153--158},
     year = {1995},
     volume = {229},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_229_a4/}
}
TY  - JOUR
AU  - L. Yu. Kolotilina
TI  - Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 1995
SP  - 153
EP  - 158
VL  - 229
UR  - http://geodesic.mathdoc.fr/item/ZNSL_1995_229_a4/
LA  - ru
ID  - ZNSL_1995_229_a4
ER  - 
%0 Journal Article
%A L. Yu. Kolotilina
%T Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality
%J Zapiski Nauchnykh Seminarov POMI
%D 1995
%P 153-158
%V 229
%U http://geodesic.mathdoc.fr/item/ZNSL_1995_229_a4/
%G ru
%F ZNSL_1995_229_a4

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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, 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.