Journal of Mathematical Sciences

, Volume 89, Issue 6, pp 1690–1693 | Cite as

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

  • L. Yu Kolotilina


Let A=(aij) i,j n =1 be a Hermitian matrix and let\(\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n \) denote its eigenvalues. If\(\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } \), 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.


Monotone Function Diagonal Entry Hermitian Matrix Hermitian Matrice Block Diagonality 
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    E. Fu and T. L. Markham, “On the eigenvalues and diagonal entries of a Hermitian matrix”,Linear Algebra Appl.,179, 7–10 (1993).CrossRefMathSciNetGoogle Scholar
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    R. A. Horn and C. R. Johnson,Topics in Matrix Analysis, Cambridge Univ. Press, New York (1991).Google Scholar
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    P. Lancaster and M. Tismenetsky,The Theory of Matrices, 2nd ed., Academic Press, New York (1985).Google Scholar
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    R. L. Smith, “Some interlacing properties of the Schur complement of a Hermitian matrix”,Linear Algebra Appl.,177, 137–144 (1992).CrossRefMATHMathSciNetGoogle Scholar

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© Plenum Publishing Corporation 1998

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  • L. Yu Kolotilina

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