Let Tn be an n × n unreduced symmetric tridiagonal matrix with eigenvalues λ1 < λ2 < ··· < λn and Wk is an (n − 1) × (n − 1) submatrix by deleting the kth row and the kth column from Tn, k = 1,2,...,n. Let μ1 ⩽ μ2 ⩽ ··· ⩽ μn−1 be the eigenvalues of Wk. It is proved that if Wk has no multiple eigenvalue, then λ1 < μ1 < λ2 < μ2 < ··· < λn−1 < μn−1 < λn; otherwise if μi = μi+1 is a multiple eigenvalue of Wk, then the above relationship still holds except that the inequality μi < λi+1 < μi+1 is replaced by μi = λi+1 = μi+1.
