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Zapiski Nauchnykh Seminarov POMI, 1995, Volume 229, Pages 153–158 (Mi znsl1714)  

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

L. Yu. Kolotilina

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract: 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.
Received: 10.05.1995
English version:
Journal of Mathematical Sciences (New York), 1998, Volume 89, Issue 6, Pages 1690–1693
DOI: https://doi.org/10.1007/BF02355372
Bibliographic databases:
UDC: 512.643
Language: Russian
Citation: L. Yu. Kolotilina, “Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality”, Computational methods and algorithms. Part XI, Zap. Nauchn. Sem. POMI, 229, POMI, St. Petersburg, 1995, 153–158; J. Math. Sci. (New York), 89:6 (1998), 1690–1693
Citation in format AMSBIB
\Bibitem{Kol95}
\by L.~Yu.~Kolotilina
\paper Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality
\inbook Computational methods and algorithms. Part~XI
\serial Zap. Nauchn. Sem. POMI
\yr 1995
\vol 229
\pages 153--158
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1714}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1433582}
\zmath{https://zbmath.org/?q=an:0899.15015|0886.15032}
\transl
\jour J. Math. Sci. (New York)
\yr 1998
\vol 89
\issue 6
\pages 1690--1693
\crossref{https://doi.org/10.1007/BF02355372}
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