An Extension of the Roots Separation Theorem

Erxiong Jiang

Annals of Operations Research, 2001, vol. 103, issue 1, 315-327

Abstract: Let T n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues λ 1 >λ 2 >⋅⋅⋅>λ n and W k is an (n−1)×(n−1) submatrix by deleting the kth row and the kth column from T n , k=1,2,...,n. Let μ 1 ≤μ 2 ≤⋅⋅⋅≤μ n−1 be the eigenvalues of W k . It is proved that if W k has no multiple eigenvalue, then λ 1 >μ 1 >λ 2 >μ 2 >⋅⋅⋅>λ n−1 >μ n−1 >λ n ; otherwise if μ i =μ i+1 is a multiple eigenvalue of W k , then the above relationship still holds except that the inequality μ i >λ i+1 >μ i+1 is replaced by μ i =λ i+1 =μ i+1 . Copyright Kluwer Academic Publishers 2001

Keywords: eigenvalue problem; symmetric tridiagonal matrix; interlace theorem; divide-and-conquer method (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1012975710842

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