An Extension of the Roots Separation Theorem

Abstract

Let T _n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues λ₁<λ₂<⋅⋅⋅<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 μ₁≤μ₂≤⋅⋅⋅≤μ _n−1 be the eigenvalues of W _k . It is proved that if W _k has no multiple eigenvalue, then λ₁<μ₁<λ₂<μ₂<⋅⋅⋅<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.