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Zapiski Nauchnykh Seminarov POMI, 1995, Volume 229, Pages 153–158
(Mi znsl1714)
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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
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
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https://www.mathnet.ru/eng/znsl1714 https://www.mathnet.ru/eng/znsl/v229/p153
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Abstract page: | 279 | Full-text PDF : | 82 |
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