Eigenvalues and diagonal elements

Rajendra Bhatia () and Rajesh Sharma ()
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Rajendra Bhatia: Ashoka University
Rajesh Sharma: H.P. University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 3, 757-759

Abstract: Abstract A basic theorem in linear algebra says that if the eigenvalues and the diagonal entries of a Hermitian matrix A are ordered as $$\lambda _{1}\le \lambda _{2}\le ...\le \lambda _{n}$$ λ 1 ≤ λ 2 ≤ . . . ≤ λ n and $$a_{1}\le a_{2}\le ...\le a_{n}$$ a 1 ≤ a 2 ≤ . . . ≤ a n , respectively, then $$\lambda _{1}\le a_{1}$$ λ 1 ≤ a 1 . We show that for some special classes of Hermitian matrices this can be extended to inequalities of the form $$\lambda _{k}\le a_{2k-1}$$ λ k ≤ a 2 k - 1 , $$k=1,2,...,\lceil \frac{n}{2}\rceil $$ k = 1 , 2 , . . . , ⌈ n 2 ⌉ .

Keywords: Hermitian matrix; Majorization; Nonnegative matrix; Laplacian matrix of graph (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00293-y

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